%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: sep02.dvi %%CreationDate: Wed Oct 02 09:44:58 2002 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips sep02 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.10.02:0944 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1000 600 600 (sep02.dvi) @start %DVIPSBitmapFont: Fa cmex10 10 1 /Fa 1 90 df89 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmcsc10 10.95 19 /Fb 19 119 df<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A2777A61D>58 D65 D70 D73 D81 D I97 D99 DII105 D107 DI110 DI<90383FC00C9038FFF81C0003EBFE3C390FE03F FC381F8007EB0003003E1301481300157C5A153CA36C141CA27E6C14006C7E13E013FE38 3FFFE06C13FE6CEBFF806C14E0000114F06C6C13F8010F13FC1300EC07FE14011400157F 153F12E0151FA37EA2151E6C143E6C143C6C147C6C14F89038C001F039FBF807E000F1B5 12C0D8E07F130038C007FC20317BAF2A>115 D<007FB712F8A39039801FF0073A7E000F E00000781678A20070163800F0163CA348161CA5C71500B3A8EC3FF8011FB512F0A32E2E 7CAD36>III E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmmi6 6 3 /Fc 3 115 df79 D<000F13FC381FC3FF3931C707803861 EC0301F813C0EAC1F0A213E03903C00780A3EC0F00EA0780A2EC1E041506D80F00130C14 3C15181538001EEB1C70EC1FE0000CEB07801F177D9526>110 D<380F01F0381FC7F838 31CE1CEA61F8EBF03C00C1137C13E014383803C000A4485AA448C7FCA4121EA2120C1617 7D951D>114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmbx12 12 22 /Fd 22 118 df46 D49 DII68 D71 D76 D80 D<903801FFE0011F13FE017F6D7E48B612E0 3A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00F090C7FCA402 03B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE00485A5B127F5B12 FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EBFFF86CECE01F C66CEB8007D90FFCC9FC322F7DAD36>97 D99 DII103 D<137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FCAAEB7F C0EA7FFFA512037EB3AFB6FCA518467CC520>105 D<90277F8007FEEC0FFCB590263FFF C090387FFF8092B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F 801FFC0003D99F009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5D A34A5DB3A7B60081B60003B512FEA5572D7CAC5E>109 D<90397F8007FEB590383FFF80 92B512E0028114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E 5CA25CA35CB3A7B60083B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139 FF001FFE000301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCAC EF7FF8A318F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CF B512F002C314C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB538 81FFE0028313F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138 E007F8ED01E092C7FCA35CB3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287 000314FF120F381FF003383FC00049133F48C7121F127E00FE140FA215077EA27F01E090 C7FC13FE387FFFF014FF6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F02 0313E0140000F0143FA26C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038 F803FE90B55A00FC5CD8F03F13E026E007FEC7FC232F7CAD2C>III E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmmi8 8 35 /Fe 35 122 df14 DI<13FC13FFEB1FC0130F6D7EA36D7EA2130180A2 6D7EA3147EA280A36E7EA2140F81A24A7E143F147FECF3F0EB01E3EB03C190380781F813 0F49C67E133E5B49137E485A48487F1207485A4848EB1F8048C7FC127E48EC0FC048EC07 E000701403232F7DAD29>21 D<0103B512F0131F137F90B612E03A01FC1F80003903F00F C03807C00748486C7E121F1300123EA25AA2140700FC5C5AA2140F5D141F92C7FC143E00 78133C147C007C5B383C01E0381F07C0D807FFC8FCEA01F8241E7D9C28>27 D<1560A315E0A25DA21401A25DA21403A292C7FCA25CEC3FE0903803FFF890380FC63E90 393E0E0F80017CEB07C03A01F00C03E0D803E0EB01F03807C01CD80F801300001F011813 F81300003E1338A2481330A2EC700100FC15F0481360150302E013E01507007801C013C0 007CEC0F800101EB1F00003C143E003E495A001F5C390F8383E03903E39F802600FFFEC7 FCEB1FF00107C8FCA21306A2130EA2130CA2131CA21318A3253C7DAD2A>30 D<160ED80380143FA20007168090C8FC000E151F001E150F001C16000018811238003013 0C141E007015061260143E023C130E00E0150C5A0238131C6C15184A1338147802F85BD8 F00114F0496C485A397C0FBE073A7FFF9FFFC0021F5B263FFC0F90C7FC391FF807FC3907 E001F0291F7F9D2C>33 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206 120E5A5A5A126009157A8714>59 D<15C0140114031580A214071500A25C140EA2141E14 1CA2143C143814781470A214F05CA213015CA213035C130791C7FCA25B130EA2131E131C A2133C1338A21378137013F05BA212015BA212035BA2120790C8FC5A120EA2121E121CA2 123C1238A212781270A212F05AA21A437CB123>61 D<1670A216F01501A24B7EA2150715 0DA2151915391531ED61FC156015C0EC0180A2EC03005C14064A7F167E5C5CA25C14E05C 4948137F91B6FC5B0106C7123FA25B131C1318491580161F5B5B120112031207000FED3F C0D8FFF8903807FFFEA22F2F7DAE35>65 D<013FB6FC17C0903A00FE0007F0EE01F84AEB 00FC177E1301177F5CA21303177E4A14FEA20107EC01FC17F84AEB03F0EE07E0010FEC1F C0EE7F009138C003FC91B55A4914FE9139C0003F804AEB0FC017E0013F140717F091C7FC 16035BA2017E1407A201FE15E0160F4915C0161F0001ED3F80EE7F004914FEED03F80003 EC0FF0B712C003FCC7FC302D7CAC35>I<92387FC003913903FFF80791391FC03E0F9139 7E00071FD901F8EB03BF4948EB01FED90FC013004948147E49C8FC017E157C49153C485A 120348481538485AA2485A173048481500A2127F90CAFCA35A5AA292381FFFFCA2923800 3FC0EE1F80163F1700A2127E5E167E7EA26C6C14FE000F4A5A6C7E6C6C1307D801F8EB1E 3CD8007EEBFC3890391FFFE018010390C8FC302F7CAD37>71 D<90273FFFFC0FB5FCA2D9 00FEC7EA3F80A24A1500A201015D177E5CA2010315FE5F5CA2010714015F5CA2010F1403 5F5C91B6FC5B9139C00007E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7 FC5BA200015D167E5BA2000315FEB539E03FFFF8A2382D7CAC3A>I<91383FFFF8A29138 007F00A2157EA215FE5DA314015DA314035DA314075DA3140F5DA3141F5DA3143FA292C7 FCA2003C5B127E00FE137E14FE5CEAFC0100F05B48485A386007E038781F80D81FFEC8FC EA07F0252E7BAC27>74 D79 D<013FB6FC17E0903A00FE0007F0EE01FC4AEB00 7EA2010181A25C1880010316005F5CA2010715FEA24A5C4C5A010F4A5A4C5A4AEB1F8004 FFC7FC91B512F84914C00280C9FCA3133F91CAFCA35B137EA313FE5BA312015BA21203B5 12E0A2312D7DAC2D>I87 D97 D<13F8121FA21201A25BA21203A25BA21207A25BA212 0FEBC7E0EB9FF8EBB83C381FF01EEBE01F13C09038800F80EA3F00A2123EA2007E131FA2 127CA2143F00FC14005AA2147EA2147C14FC5C387801F01303495A383C0F806C48C7FCEA 0FFCEA03F0192F7DAD1E>II<151FEC03FFA2EC003F A2153EA2157EA2157CA215FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F039 03E007E0EA07C0120FEA1F8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F02 1506143E5AEC7E0E007CEBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903 F003C0202F7DAD24>II<157C4AB4FC913807C380EC0F87150FEC1F1FA391383E0E0092C7FCA3147E147CA414 FC90383FFFF8A2D900F8C7FCA313015CA413035CA413075CA5130F5CA4131F91C8FCA413 3EA3EA383C12FC5BA25B12F0EAE1E0EA7FC0001FC9FC213D7CAE22>I<14FCEB03FF9038 0F839C90381F01BC013E13FCEB7C005B1201485A15F8485A1401120F01C013F0A2140312 1F018013E0A21407A215C0A2000F130F141F0007EB3F80EBC07F3803E1FF3800FF9F9038 3E1F0013005CA2143EA2147E0038137C00FC13FC5C495A38F807E038F00F80D87FFEC7FC EA1FF81E2C7E9D22>I<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA 0E3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC0 4014C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18> 105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813 C01330EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA2 5CA21301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F80 1C3B81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F890 38F003FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0F DC13F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEB F038ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<27078007F0137E3C 1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01263079C001B87F26707F8013B000 60010013F001FE14E000E015C0485A4914800081021F130300015F491400A200034A1307 6049133E170F0007027EEC8080188149017C131F1801000F02FCEB3F03053E130049495C 180E001F0101EC1E0C183C010049EB0FF0000E6D48EB03E0391F7E9D3E>109 D<3907C007E0391FE03FF83918F8783E393879E01E39307B801F38707F00126013FEEAE0 FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF8081618EBC00115F0000F 1538913803E0300180147016E0001F010113C015E390C7EAFF00000E143E251F7E9D2B> I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C0000713F000069038E0 03E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C0133EA2017EEB1F80A2 017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FFC7FCEBF0FC000390 C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<3807C01F390FF07FC039 1CF8E0E0383879C138307B8738707F07EA607E13FC00E0EB03804848C7FCA2128112015B A21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B1F7E9D20>114 DI118 DI<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F6 0F000E13FE4813FC12180038EC0700003049C7FCA2EA200100005BA313035CA301075B5D 14C000385CD87C0F130600FC140E011F130C011B131C39F03BE038D8707113F0393FE0FF C0260F803FC7FC221F7E9D28>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy8 8 4 /Ff 4 49 df0 D<123C127E12FFA4127E123C08087A9414>I<13 0C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FFE0 38007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC039781E 078000601301000090C7FCA5130C1A1D7C9E23>3 D<137813FE1201A3120313FCA3EA07 F8A313F0A2EA0FE0A313C0121F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413> 48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr9 9 25 /Fg 25 122 df45 D48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB5 1280A319327AB126>II52 D56 DI67 D 70 D78 D82 D<90381FE00390387FFC0748B5FC3907F01FCF390F8003FF48C7FC003E8081 4880A200788000F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F06C13FF6C14C06C 14F0C680013F7F01037F9038003FFF140302001380157F153FED1FC0150F12C0A21507A3 7EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B512E0011F138026 C003FEC7FC22377CB42B>I97 DI<153FEC0FFFA3EC 007F81AEEB07F0EB3FFCEBFC0F3901F003BF3907E001FF48487E48487F8148C7FCA25A12 7E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C4813803A03F007BFFC3900F81E3FEB3FFC D90FE0130026357DB32B>100 DI<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F83809039 800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380EFFF8 380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F0001003EEB 007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6B512 80D91FFCC7FC22337EA126>103 D105 D<3903F01FC000FFEB7FF090 38F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326 217EA02B>110 DI<3903F0 3F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07 F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F890 38F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<3803E07C38FFE1FF9038 E38F809038E71FC0EA07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA0 1E>114 D<1330A51370A313F0A21201A212031207381FFFFEB5FCA23803F000AF1403A8 14073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>116 DI<3A7FFF807FF8A33A07F8001FC00003 EC0F800001EC070015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E001 0F5BA2903807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05C A2495A1278D8FC03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr6 6 4 /Fh 4 51 df<130C1338137013E0EA01C0EA038013005A120EA25AA25AA312781270A312 F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013701338130C0E317AA418> 40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338A3133CAB1338A3 13781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I<13E01201 120712FF12F91201B3A7487EB512C0A212217AA01E>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmti10 10.95 63 /Fi 63 125 df<933807FF80043F13E09338FE00F8DB01F0133EDB07E0130E4B48131F4C 137F031F14FF4BC7FCA218FE157E1878180015FE5DA31401A25DA414030103B712F0A218 E0903A0003F000070207140F4B14C0A3171F020F15805DA2173F1800141F5D5F177EA214 3F92C712FE5FA34A1301027EECF81CA3160302FEECF03C4A1538A21878187013014A0101 13F018E0933800F1C0EF7F804948EC1F0094C7FCA35C1307A2001E5B127F130F00FF5BA2 49CAFC12FEEAF81EEA703CEA7878EA1FF0EA07C0385383BF33>12 DII39 DI<140315 80A2EC01C0EC00E0A21570A215781538153CA3151EA4151FA2150FA7151FA9153FA2153E A3157EA2157CA215FCA215F8A21401A215F0A2140315E0A2140715C0A2EC0F80A2141F15 005C143EA25CA25CA2495A5C1303495A5C130F49C7FC131E5B137C5B5B485A485A485A48 C8FC121E5A12705A5A205A7FC325>I44 D<387FFFFEA3B5FCA21705799521>I<120FEA3FC0127FA212FFA31380EA7F0012 3C0A0A77891C>I<15FE913807FF8091381F07C091387C01F0ECF000494813F849481378 0107147C495A49C7FC167E133E137EA25BA2485AA2000315FEA25B000715FCA249130112 0FA34848EB03F8A44848EB07F0A448C7EA0FE0A316C0007E141F12FE1680153FA2481500 A2157EA25DA25D4813015D6C495A127C4A5A4A5A6C49C7FC143E6C5B380FC1F03803FFC0 C648C8FC273F76BC2E>48 D<15031507150F151F151E153E157EEC01FEEC03FC1407141F EB01FF90380FFBF8EB1FC3EB0E07130015F0A2140FA215E0A2141FA215C0A2143FA21580 A2147FA21500A25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CEB 7FE0B612F0A215E0203D77BC2E>I<15FE913803FFC091380F01F091383C00F84A137C4A 7F4948133F49487F4A148049C7FC5BEB0E0C011E15C0EB1C0EEB3C06133813781370020E 133FD9F00C148013E0141C0218137F00011600EBC0384A13FEEC600102E05B3A00E3C003 F89039FF0007F0013C495A90C7485A5E037FC7FC15FC4A5A4A5AEC0FC04AC8FC147E14F8 EB03E0495A011FC9FC133E49141801F0143C48481438485A1678485A48C85A120E001E4A 5AD83FE0130301FF495A397C3FF01FD8780FB55AD8700391C7FCD8F0015B486C6C5A6E5A EC07C02A3F79BC2E>II<02C0EB018002F0130FD901FEEB7F0091B512FE5E5E4914E016804B C7FCECBFF8D90780C8FC91C9FCA35B130EA3131E131CA3133C9038381FC0ECFFF090383B E07C90387F003E017E133F017C7F0178805B498090C7FCA6153FA4001F147F486C5C487E A24913FF00FF92C7FC90C7FC48495A12E04A5A5D6C495A140F00705C0078495A6C495A00 3E01FEC8FC381F03FC380FFFF0000313C0C648C9FC293F77BC2E>53 DI<131E EB3F80137FEBFFC05AA214806C13005B133C90C7FCB3120FEA3FC0127FA212FFA35B6CC7 FC123C122777A61C>58 D<171C173C177CA217FCA216011603A21607A24C7EA2161DA216 391679167116E1A2ED01C1A2ED038115071601150EA2031C7FA24B7EA25D15F05D4A5AA2 4A5AA24AC7FC5C140E5C021FB6FC4A81A20270C7127FA25C13015C495AA249C8FCA2130E 131E131C133C5B01F882487ED807FEEC01FFB500E0017FEBFF80A25C39417BC044>65 D<9339FF8001C0030F13E0033F9038F803809239FF807E07913A03FC001F0FDA0FF0EB07 1FDA1FC0ECBF00DA7F806DB4FC4AC77E495AD903F86E5A495A130F4948157E4948157C49 5A13FF91C9FC4848167812035B1207491670120FA2485A95C7FC485AA3127F5BA312FF5B A490CCFCA2170FA2170EA2171E171C173C173817786C16706D15F04C5A003F5E6D140300 1F4B5A6D4AC8FC000F151E6C6C5C6C6C14F86C6C495A6C6CEB07C090397FC03F8090261F FFFEC9FC010713F0010013803A4272BF41>67 D<49B812F8A390260003FEC7121F18074B 14031801F000F014075DA3140F5D19E0A2141F4B1338A2EF7801023F027013C04B91C7FC A217F0027F5CED80011603160F91B65AA3ED001F49EC07805CA3010392C8FC5CF003804C 13070107020E14005C93C75A180E010F161E4A151C183CA2011F5E5C60A2013F15014A4A 5A1707017F150F4D5A4A147F01FF913807FF80B9FCA295C7FC3D3E7BBD3E>69 D<49B812F0A390260003FEC7123F180F4B1403A2F001E014075DA3140F5D19C0A2141F5D 1770EFF003023F02E013804B91C7FCA21601027F5CED8003A2160702FFEB1F8092B5FCA3 49D9003FC8FC4A7F82A20103140E5CA2161E0107141C5CA293C9FC130F5CA3131F5CA313 3F5CA2137FA25C497EB612E0A33C3E7BBD3B>II<49B648B6FC495DA2D9000390C7000313004B5D4B5DA2180714074B5DA2180F140F4B 5DA2181F141F4B5DA2183F143F4B5DA2187F147F4B5DA218FF91B8FC96C7FCA292C71201 5B4A5DA2170313034A5DA2170713074A5DA2170F130F4A5DA2171F131F4A5DA2173F133F 4A5DA2017F157FA24A5D496C4A7EB66CB67EA3483E7BBD44>I<49B6FC5BA2D900031300 5D5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C7FCA35B5CA313035CA3 13075CA3130F5CA3131F5CA3133F5CA2137FA25C497EB67EA3283E7BBD23>I<4AB61280 A2180091C713C0167F5FA216FF94C7FCA35D5EA315035EA315075EA3150F5EA3151F5EA3 153F5EA3157FA25EA215FFA293C8FCA25CA25DA2380F8003EA3FC0D87FE05BA21407D8FF C05B140F01805B49485A12FC0070495A4A5A6C01FEC9FC383C01FC380F07F03807FFC0C6 48CAFC314079BD30>I<49B6903807FFFE605ED9000390C7000113E04B6E13004B15FC4E 5A19E002074B5A4BEC0F804EC7FC183C020F5D4B5C4D5AEF07C0021F4AC8FC4B131E5F5F 023F5C9238C003E0EE07804CC9FC027F5B4B5AEEFF801581ECFF834B7FED0F7FED1E3F49 017C7FECFEF89138FFE01F03C07F491380ED000F4A805C010714074A80A21603010F815C 160183131F4A6D7FA2177F013F825C173F017F82A24A81496C4A7EB6D8800FB512C0A261 473E7BBD46>I<49B612C0A25FD9000390C8FC5D5DA314075DA3140F5DA3141F5DA3143F 5DA3147F5DA314FF92C9FCA35B5CA313035C18C0EF01E0010716C05C17031880130F4A14 0718005F131F4A141EA2173E013F5D4A14FC1601017F4A5A16074A131F01FFECFFF0B8FC A25F333E7BBD39>I<49B5933807FFFC496062D90003F0FC00505ADBBF805E1A771AEF14 07033F923801CFE0A2F1039F020FEE071F020E606F6C140E1A3F021E161C021C04385BA2 F1707F143C023804E090C7FCF001C0629126780FE0495A02705FF00700F00E0114F002E0 031C5BA2F03803010116704A6C6C5D18E019070103ED01C00280DA03805BA2943807000F 13070200020E5C5FDB03F8141F495D010E4B5CA24D133F131E011CDAF9C05CEEFB80197F 013C6DB4C7FC013895C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F04948B512FE16 E01500563E7BBD52>I<902601FFFE020FB5FC496D5CA2D900016D010013C04AEE3F0019 3E70141C193CEC07BFDB3FE01438151F1978020F7FDA0E0F15708219F0EC1E07021C6D5C A203031401023C7FDA38015DA2701303EC7800027002805BA2047F130702F014C04A013F 91C7FCA2715A0101141F4AECF00EA2040F131E010315F84A151C1607EFFC3C0107140391 C7143817FE040113784915FF010E16708218F0131E011C6F5AA2173F133C01385E171F13 7813F8486C6F5AEA07FEB500F01407A295C8FC483E7BBD44>II<49B77E18F018FC 903B0003FE0003FEEF00FF4BEC7F80F03FC00207151F19E05DA2020F16F0A25DA2141FF0 3FE05DA2023F16C0187F4B1580A2027FEDFF00604B495A4D5A02FF4A5A4D5A92C7EA3FC0 4CB4C7FC4990B512FC17E04ACAFCA21303A25CA21307A25CA2130FA25CA2131FA25CA213 3FA25CA2137FA25C497EB67EA33C3E7BBD3E>I<49B612FCEFFF8018F0903B0003FE000F F8EF03FE4BEB00FF8419800207ED3FC05DA219E0140F5DA3021FED7FC05DA2F0FF80143F 4B15004D5A60027F4A5A4B495A4D5AEF3F8002FF02FEC7FC92380007F892B512E0178049 9038000FE04A6D7E707E707E0103814A130083A213075CA25E130F5C5F1603131F5CA301 3F020714404A16E05F017F160119C04A01031303496C1680B6D8800113079438FE0F0093 38007E1ECAEA3FFCEF07F03B407BBD42>82 D<92390FF001C0ED7FFE4AB5EA0380913907 F80FC791390FC003EF91391F8001FF4AC71300027E805C495A4948143EA2495AA2010F15 3C5CA3011F1538A38094C7FC80A214FC6DB4FC15F015FE6DEBFFC06D14F06D14FC6D8014 3F020F7F020180EC001F150303007F167F163FA2161FA212075A5F120EA2001E153F94C7 FCA2163E003E157E167C003F15FC4B5A486C5C4B5A6D495AD87DE0EB1F80D8F8F849C8FC 017F13FE39F03FFFF8D8E00F13E048C690C9FC32427ABF33>I<48B9FCA25A903AFE001F F00101F89138E0007FD807E0163E49013F141E5B48C75BA2001E147FA2001C4B131C123C 003814FFA2007892C7FC12704A153C00F01738485CC716001403A25DA21407A25DA2140F A25DA2141FA25DA2143FA25DA2147FA25DA214FFA292C9FCA25BA25CA21303A25CEB0FFE 003FB67E5AA2383D71BC41>I86 D<277FFFFE01B500FC90B512E0B5FCA20003902680000790C7380FFC006C90C701 FCEC07F049725A04035EA26350C7FCA20407150EA2040F5D1A3C041F153862163B621673 4F5A6D14E303014B5A6C15C303034BC8FC1683DB0703140E191E030E151C61031C7F61ED 380161157003F04A5A15E002014B5A15C0DA03804AC9FC60DA0700140E60140E605C029C 5D14B8D97FF85D5C715A5C4A5DA24A92CAFC5F91C7FC705A137E5F137C5F137801705D53 406EBD5B>I<027FB612FEA3913AFFF80007FC03C014F892C7EA0FF0D901FC141F4AEC3F E04AEC7FC04A15800103EDFF004A5B4C5A4948495A5F91C7485A49141F010E4A5A4C5A5F 011E4AC7FC90C75A4B5A4B5A5E4B5A151F4B5A4B5A5E15FF4A90C8FC4A5A4A5A5D140F4A 5A4A5A4A48130E4B131E02FF141C4990C7FC495A4948143C4A1438010F1578495A494814 70494814F05C01FF4A5A4890C7FC4848140348481407494A5A000F151F4848143F4848EC FF804848130F90B7FCB8FC94C7FC373E79BD38>90 D<147E49B47E903907C1C38090391F 80EFC090383F00FF017E137F4914804848133F485AA248481400120F5B001F5C157E485A A215FE007F5C90C7FCA21401485C5AA21403EDF0385AA21407EDE078020F1370127C021F 13F0007E013F13E0003E137FECF3E1261F01E313C03A0F8781E3803A03FF00FF00D800FC 133E252977A72E>97 DIII< EC3F80903801FFE0903807E0F890381F803CEB3E0001FC131E485A485A12074848133E49 133C121F4848137C15F8EC03F0397F000FE0ECFF80B5EAFC0014C048C8FCA45AA6150615 0E151E007C143C15786C14F0EC01E06CEB07C0390F801F003807C0FC3801FFF038007F80 1F2976A72A>I<167C4BB4FC923807C78092380F83C0ED1F87161FED3F3FA2157EA21780 EE0E004BC7FCA414015DA414035DA30103B512F8A390260007E0C7FCA3140F5DA5141F5D A4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495AA4495A121E127F5C12FF49C9 FCA2EAFE1EEAF83C1270EA7878EA3FE0EA0F802A5383BF1C>III<1478EB01FCA21303A314F8EB00E01400 AD137C48B4FC38038F80EA0707000E13C0121E121CEA3C0F1238A2EA781F00701380A2EA F03F140012005B137E13FE5BA212015BA212035B1438120713E0000F1378EBC070A214F0 EB80E0A2EB81C01383148038078700EA03FEEA00F8163E79BC1C>I<1507ED1FC0A2153F A31680ED0E0092C7FCADEC07C0EC3FF0EC78F8ECE07CEB01C01303EC807EEB0700A2010E 13FE5D131E131CEB3C01A201005BA21403A25DA21407A25DA2140FA25DA2141FA25DA214 3FA292C7FCA25CA2147EA214FEA25CA213015CA2121C387F03F012FF495A5C495A4848C8 FCEAF83EEA707CEA3FF0EA0FC0225083BC1C>IIIIII<903903E001F890390FF8 07FE903A1E7C1E0F80903A1C3E3C07C0013C137801389038E003E0EB783F017001C013F0 ED80019038F07F0001E015F8147E1603000113FEA2C75AA20101140717F05CA20103140F 17E05CA20107EC1FC0A24A1480163F010F15005E167E5E131F4B5A6E485A4B5A90393FB8 0F80DA9C1FC7FCEC0FFCEC03E049C9FCA2137EA213FEA25BA21201A25BA21203A2387FFF E0B5FCA22D3A80A72E>I<027E1360903901FF81E0903807C1C390391F80E7C090383F00 F7017E137F5B4848EB3F80485AA2485A000F15005B121F5D4848137EA3007F14FE90C75A A3481301485CA31403485CA314074A5A127C141F007E133F003E495A14FF381F01EF380F 879F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5CA21301130390B512F05AA2 233A77A72A>IIII<137C48B4141C26038F8013 7EEA0707000E7F001E15FE121CD83C0F5C12381501EA781F007001805BA2D8F03F130314 0000005D5B017E1307A201FE5C5B150F1201495CA2151F0003EDC1C0491481A2153F1683 EE0380A2ED7F07000102FF13005C01F8EBDF0F00009038079F0E90397C0F0F1C90391FFC 07F8903907F001F02A2979A731>I<017CEB01C048B4EB07F038038F80EA0707000E01C0 13F8121E001C1403EA3C0F0038EC01F0A2D8781F130000705BA2EAF03F91C712E012005B 017E130116C013FE5B1503000115805BA2ED07001203495B150EA25DA25D157800011470 6D5B0000495A6D485AD97E0FC7FCEB1FFEEB03F0252979A72A>I<017C167048B4913870 01FC3A038F8001F8EA0707000E01C015FE001E1403001CEDF000EA3C0F0038177C1507D8 781F4A133C00701380A2D8F03F130F020049133812005B017E011F14784C137013FE5B03 3F14F0000192C712E05BA2170100034A14C049137E17031880A2EF070015FE170E000101 01141E01F86D131C0000D9039F5BD9FC076D5A903A3E0F07C1E0903A1FFC03FFC0902703 F0007FC7FC372979A73C>I<903903F001F890390FFC07FE90393C1E0E0F9026780F1C13 8001F0EBB83FD801E013F89039C007F07FEA0380000714E0D9000F140048151C000E4AC7 FCA2001E131FA2C75BA2143F92C8FCA35C147EA314FE4A131CA30101143C001E1538003F 491378D87F811470018314F000FF5D9039077801C039FE0F7C033A7C0E3C078027783C1E 1EC7FC391FF80FFC3907E003F029297CA72A>I<137C48B4143826038F8013FCEA070700 0E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F1307020013E0 12005B017E130F16C013FE5B151F1201491480A2153F000315005BA25D157EA315FE5D00 011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA21407000E5CEA3F80007F 495AA24A5AD8FF0090C7FC143E007C137E00705B387801F0383803E0381E0FC06CB4C8FC EA03F8263B79A72C>II124 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi10 10.95 59 /Fj 59 122 df11 D14 DI<133F14E0EB07F0EB03FC 13016D7EA3147FA26E7EA36E7EA36E7EA36E7EA36E7EA26E7EA36E7EA3157FA36F7E157F 15FF4A7F5C913807CFE0EC0F8FEC1F0F91383E07F0147C14FC49486C7EEB03F0EB07E049 486C7EEB1F80EB3F00496D7E13FE4848147F485A485A4848EC3F80485A123F4848EC1FC0 48C8FC4816E048150F48ED07F0007015032C407BBE35>21 D<020FB512FE027F14FF49B7 FC1307011F15FE903A3FE03FE00090387F000F01FE6D7E4848130348488048481301485A 5B121F5B123F90C7FC5A127EA2150300FE5D5AA24B5AA2150F5E4B5AA2007C4AC7FC157E 157C6C5C001E495A001FEB07E0390F800F802603E07EC8FC3800FFF8EB3FC030287DA634 >27 D<011FB612C090B7FC5A5A481680260FC007C8FC48C65A123E003C130E48131E5A5A A2C75AA3147CA2147814F8A4495AA31303A25CA21307A3495AA3131FA25C6DC9FC2A287D A628>I<1678A21670A216F0A25EA21501A25EA21503A25EA21507A293C7FCA25DA2150E EDFFC0020F13FC91383F9E3F903A01F81C0FC0D903E0EB03E0903A0FC03C01F0D91F00EB 00F8017E0138137C5B48480178133E485A48480170133F120F4901F0131F485A5D48C7FC 0201143F5A007E5CA20203147F00FE167E485C17FE020714FC1601007C020013F8EE03F0 007E49EB07E0A2003E010EEB0FC0003FED1F806C011EEB3F00D80F80147C3A07C01C01F8 D803E0EB03E03A01F03C1F80D8007E01FEC7FC90381FFFF801011380D90078C8FCA21470 A214F0A25CA21301A25CA21303A25CA21307A230527CBE36>30 D32 D<012016E00178ED03F801F8ED07FC12015B485A5B00 07160349150148CAFC187C121EA2001C173C003C021C14380038147EA20078177803FE14 7000705CA218F04B14E000F001011401A24B14C017034B1307020315806C496C130FEF1F 006CD91FF85B007C496C137E007E496C5B3A7F83FE7F87263FFFFCEBFFF84A6C5B6C01F0 5C6CD9C01F5B6CD9000790C7FCD801FCEB01F836297FA739>I<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A798919>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A4 13E013C0A312011380120313005A120E5A1218123812300B1C798919>I<180E183F18FF EF03FEEF0FF8EF3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED1FF8ED7F E0913801FF80DA07FEC8FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07 FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF 9038007FC0EC1FF0EC07FE913801FF809138007FE0ED1FF8ED03FE923800FF80EE3FE0EE 0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FF183F180E383679B147>II< 127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0 EC1FF8EC07FE913801FF809138007FE0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE 933800FF80EF3FE0EF0FF8EF03FEEF00FFA2EF03FEEF0FF8EF3FE0EFFF80933803FE00EE 0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1FF8EC7FC049 48C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270383679B1 47>I<17075F84171FA2173F177FA217FFA25E5EA24C6C7EA2EE0E3F161E161C1638A216 70A216E0ED01C084ED0380171FED07005D150E5DA25D157815705D844A5A170F4A5A4AC7 FC92B6FC5CA2021CC7120F143C14384A81A24A140713015C495AA249C8FC5B130E131E49 82137C13FED807FFED1FFEB500F00107B512FCA219F83E417DC044>65 D<49B712F818FF19E090260001FEC7EA3FF0F007F84B6E7E727E850203815D1A80A20207 167F4B15FFA3020F17004B5C611803021F5E4B4A5A180FF01FE0023F4B5A4B4A5ADD01FE C7FCEF07F8027FEC7FE092B6C8FC18E092C7EA07F84AEC01FE4A6E7E727E727E13014A82 181FA213034A82A301075F4A153FA261010F167F4A5E18FF4D90C7FC011F5E4A14034D5A 013FED1FF04D5A4AECFFC0017F020790C8FCB812FC17F094C9FC413E7DBD45>II<49B712F818FF19C0D9000190C7EA3FF0 F00FF84BEC03FCF000FE197F0203EE3F805DF11FC0A20207EE0FE05D1AF0A2020F16075D A21AF8141F5DA2190F143F5DA21AF0147F4B151FA302FF17E092C9123FA21AC049177F5C 1A8019FF010318005C4E5A61010716034A5E4E5A180F010F4C5A4A5E4E5A4EC7FC011F16 FE4A4A5AEF07F8013FED0FE0EF3FC04A49B4C8FC017FEC0FFCB812F017C004FCC9FC453E 7DBD4B>I<49B912C0A3D9000190C71201F0003F4B151F190F1A80020316075DA314075D 1A00A2140F4BEB0380A205075B021FED000E4B92C7FC5FA2023F141E5D173EEE01FE4AB5 5AA3ED800102FF6D5A92C71278A34915705C191C05F0133C01034B13384A167894C71270 A2010717F04A5E180161010F16034A4B5AA2180F011F4CC7FC4A5D187E013F16FE4D5A4A 140F017F15FFB95AA260423E7DBD43>I<49B9FCA3D9000190C7120718004B157F193F19 1E14035DA314075D191CA2140F5D17074D133C021F020E13384B1500A2171E023F141C4B 133C177C17FC027FEB03F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101035D 5CA293C9FC13075CA3130F5CA3131F5CA2133FA25C497EB612F8A3403E7DBD3A>II<49B6D8C03FB5 12F81BF01780D900010180C7383FF00093C85B4B5EA2197F14034B5EA219FF14074B93C7 FCA260140F4B5DA21803141F4B5DA21807143F4B5DA2180F4AB7FC61A20380C7121F14FF 92C85BA2183F5B4A5EA2187F13034A5EA218FF13074A93C8FCA25F130F4A5DA21703131F 4A5DA2013F1507A24A5D496C4A7EB6D8E01FB512FCA2614D3E7DBD4C>I<49B612C05BA2 D90001EB800093C7FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314 FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612E0A32A 3E7DBD28>I<92B612E0A39239003FF000161F5FA2163F5FA3167F5FA316FF94C7FCA35D 5EA315035EA315075EA3150F5EA3151FA25EA2153FA25EA2157FA25EA2D80F8013FFEA3F C0486C91C8FCA25CD8FFC05B140301805B49485A00FC5C0070495A0078495A0038495A00 1E017EC9FC380F81FC3803FFE0C690CAFC33407ABD32>I<49B600C090387FFFF896B5FC 5FD900010180C7000F130093C813F84B16E01A804FC7FC0203163C4B15F84E5AF003C002 074B5A4B021FC8FC183E1878020F5D4BEB03E0EF07804DC9FC021F143E4B5B17F04C5A02 3F1307EDC00F4C7E163F027FEBFFF8ED81EFED83CF92388F87FC9138FF9F0792383C03FE 15784B6C7E4913E0158092C77F5C01036F7E5C717EA213074A6E7EA2717E130F4A6E7EA2 84011F15035C717E133F855C496C4A13E0B600E0017F13FFA34D3E7DBD4D>I<49B612F0 A3D900010180C7FC93C8FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5D A314FF92C9FCA35B5C180C181E0103161C5C183C183813074A1578187018F0130F4AEC01 E0A21703011FED07C04A140F171F013FED3F8017FF4A1303017F021F1300B9FCA25F373E 7DBD3E>I<49B56C93383FFFF05113E098B5FCD90001F1E000704B5B03DF933803BF80A2 F2077F1403039F040E90C7FC1A1CDB8FE05E02075F030F4C5AA21AE1020FEE01C1020E60 6F6CEC03811A83021EEE0703021C040E5BA2F11C07023C16380238606F6C1470F1E00F14 780270DB01C05BA2953803801F02F0ED07004A6C6C5E180E4E133F130102C04B5C601A7F 01036D6C5B4A95C8FC4D5A4D485B130791C749C75A170E047F1401495D010E4B5CA24D13 03131E011C4B5C5F013C023F1407017C5D01FE92C75BD803FF4D7EB500FC013E011FB512 F8163C041C5E5C3E7DBD58>I<49B56C49B512F81BF0A290C76D9039000FFE004AEE03F0 705D735A03DF150302037F038F5E82190791380787FC030793C7FC1503705C140F91260E 01FF140EA26F151E021E80021C017F141C83193C023C6D7E02381638161F711378147802 706D6C1370A2040714F002F0804A01035C8318010101EC01FF4A5E82188313034A91387F C380A2EF3FC7010716E791C8001F90C8FC18F718FF4981010E5E1707A2131E011C6F5AA2 013C1501137C01FE6F5AEA03FFB512FC187818704D3E7DBD49>I<49B712F018FF19C0D9 000190C76C7EF00FF84BEC03FC1801020382727E5DA214071A805DA2140F4E13005DA202 1F5E18034B5D1807023F5E4E5A4B4A5A4E5A027F4B5A06FEC7FC4BEB03FCEF3FF091B712 C005FCC8FC92CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA2 5C497EB612E0A3413E7DBD3A>80 DI<49B77E18 F818FFD90001D900017F9438003FE04BEC0FF0727E727E14034B6E7EA30207825DA3020F 4B5A5DA24E5A141F4B4A5A614E5A023F4B5A4B4A5A06FEC7FCEF03FC027FEC0FF04BEBFF 8092B500FCC8FC5F9139FF8001FE92C7EA7F80EF1FC084496F7E4A1407A28413035CA217 0F13075C60171F130F5CA3011F033F5B4AEE038018E0013F17071A004A021F5B496C160E B600E090380FF01E05075B716C5ACBEAFFE0F03F8041407DBD45>II<007FB500F090387FFFFE19FC5D26007FE0C7000313804A913800FC 004A5D187001FF16F0A291C95AA2481601605BA200031603605BA20007160795C7FC5BA2 000F5E170E5BA2001F161E171C5BA2003F163C17385BA2007F1678A2491570A200FF16F0 A290C95AA216015F5A16035F16074CC8FC160E161E5E007F5D5E6C4A5A6D495A6C6C495A 6C6C011FC9FC6C6C137E3903FC03F8C6B512E0013F1380D907FCCAFC3F407ABD3E>85 DII<027FB5D88007B512C091B6FCA2020101F8C7EBF8009126007FE0EC7F804C92C7 FC033F157C701478616F6C495A4E5A6F6C495A4EC8FC180E6F6C5B606F6C5B6017016F6C 485A4D5A6F018FC9FC179E17BCEE7FF85F705AA3707EA283163F167FEEF7FCED01E7EEC3 FEED0383ED070392380E01FF151E4B6C7F5D5D4A486D7E4A5A4A486D7E92C7FC140E4A6E 7E5C4A6E7E14F0495A49486E7E1307D91F806E7ED97FC014072603FFE0EC1FFF007F01FC 49B512FEB55CA24A3E7EBD4B>II97 DIIII<163EEEFFC0923803E1E09238 07C0F0ED0F811687ED1F8F160F153FA217E092387E038093C7FCA45DA514015DA30103B5 12FCA390260003F0C7FCA314075DA4140F5DA5141F5DA4143F92C8FCA45C147EA414FE5C A413015CA4495AA35CEA1E07127F5C12FF495AA200FE90C9FCEAF81EEA703EEA7878EA1F F0EA07C02C537CBF2D>I II<143C14FEA21301A314FCEB00701400AD137E3801FF803803C7C0 EA0703000F13E0120E121C13071238A2EA780F007013C0A2EAF01F14801200133F14005B 137EA213FE5BA212015B0003130E13F0A20007131EEBE01CA2143CEBC0381478147014E0 13C13803E3C03801FF00EA007C173E7EBC1F>III109 DI112 D<91381F800C9138FFE01C903903F0707C90390FC0387890391F801CF890383F 000F137E4914F000011407485A485A16E0485A121F150F484814C0A3007F141F491480A3 00FF143F90C71300A35D48147EA315FE007E495A1403A26C13074A5A381F801D000F1379 3807C1F33901FFC3F038007F03130014075DA3140F5DA3141F5DA2143F147F90381FFFFE 5BA2263A7DA729>III<137C48B4EC03802603C7C0EB0F C0EA0703000F7F000E151F121C010715801238163FEA780F0070491400A2D8F01F5C5C00 00157E133F91C712FEA2495C137E150113FE495CA215030001161C4914F0A21507173CEE E038150F031F1378000016706D133F017C017313F0017E01E313E0903A3F03C1F1C0903A 0FFF007F80D901FCEB1F002E297EA734>117 D<017E147848B4EB01FC2603C7C013FED8 07031303000F13E0120E121C0107130100381400167ED8780F143E00705B161EEAF01F4A 131C1200133F91C7123C16385B137E167801FE14705B16F016E0120149EB01C0A2ED0380 A2ED0700A20000140E5D6D133C017C5B6D5B90381F03C0903807FF80D901FCC7FC27297E A72C>I<017CEE038048B40207EB0FE02603C7C090391F801FF0EA0703000F7F000E153F 001C16000107160F003817074C1303D8780F027E130100705B1800D8F01F14FE4A4914E0 1200133FDA000114014C14C05B137E0303140301FE4A14805BA2F0070000011407494A5B 180EA260A2030F5C12006D011F5C017C496C5B017E0139495A6D903870F80390281F81E0 7C0FC7FC903A07FFC01FFE010090380007F03C297EA741>II<137C48B4 EC03802603C7C0EB0FC0EA0703000F7F000E151F001C168013071238163FD8780F150000 705BA2D8F01F5C4A137E1200133F91C712FE5E5B137E150113FE495CA2150300015D5BA2 15075EA2150F151F00005D6D133F017C137F017E13FF90393F03DF8090380FFF1FEB01FC 90C7123F93C7FCA25DD80380137ED80FE013FE001F5C4A5AA24848485A4A5A6CC6485A00 1C495A001E49C8FC000E137C380781F03803FFC0C648C9FC2A3B7EA72D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmbx12 14.4 37 /Fk 37 123 df<157815FC14031407141F14FF130F0007B5FCB6FCA2147F13F0EAF800C7 FCB3B3B3A6007FB712FEA52F4E76CD43>49 DI<9138 0FFFC091B512FC0107ECFF80011F15E090263FF8077F9026FF800113FC4848C76C7ED803 F86E7E491680D807FC8048B416C080486D15E0A4805CA36C17C06C5B6C90C75AD801FC16 80C9FC4C13005FA24C5A4B5B4B5B4B13C04B5BDBFFFEC7FC91B512F816E016FCEEFF80DA 000713E0030113F89238007FFE707E7013807013C018E07013F0A218F8A27013FCA218FE A2EA03E0EA0FF8487E487E487EB57EA318FCA25E18F891C7FC6C17F0495C6C4816E001F0 4A13C06C484A1380D80FF84A13006CB44A5A6CD9F0075BC690B612F06D5D011F15800103 02FCC7FCD9001F1380374F7ACD43>I<177C17FEA2160116031607160FA2161F163F167F A216FF5D5DA25D5DED1FBFED3F3F153E157C15FCEC01F815F0EC03E01407EC0FC01580EC 1F005C147E147C5C1301495A495A5C495A131F49C7FC133E5B13FC485A5B485A1207485A 485A90C8FC123E127E5ABA12C0A5C96C48C7FCAF020FB712C0A53A4F7CCE43>III<91B5FC010F14F8017F14FF90B712C00003D9C00F7F2707 FC00017FD80FE06D7F48486E7E48C87FD87FE06E7E7F7F486C1680A66C5A18006C485C6C 5AC9485A5F4B5B4B5B4B5B4B5B4B90C7FC16FC4B5A4B5A16C04B5A93C8FC4A5A5D14035D 5D14075DA25D140FA25DAB91CAFCAAEC1FC04A7EECFFF8497FA2497FA76D5BA26D5BEC3F E06E5A315479D340>63 D<171F4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C7FA24C7FA3 4C8083047F80167E8304FE804C7E03018116F8830303814C7E03078116E083030F814C7E 031F81168083033F8293C77E4B82157E8403FE824B800201835D840203834B800207835D 844AB87EA24A83A3DA3F80C88092C97E4A84A2027E8202FE844A82010185A24A82010385 4A82010785A24A82010F855C011F717FEBFFFCB600F8020FB712E0A55B547BD366>65 D<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA803F0203 DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848C9FC4901 F8824949824949824949824949824990CA7E494883A2484983485B1B7F485B481A3FA248 49181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C7FA21B0F6C6D19 80A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D6D4C5A6D6D4C5A 6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1FF80203903AFFE0 01FFF0020091B612C0033F93C8FC030715FCDB007F14E0040101FCC9FC525479D261>67 D73 D75 D<93380FFFC00303B6FC031F15E092B712FC 0203D9FC0013FF020F01C0010F13C0023F90C7000313F0DA7FFC02007F494848ED7FFE49 01E0ED1FFF49496F7F49496F7F4990C96C7F49854948707F4948707FA24849717E48864A 83481B804A83481BC0A2481BE04A83A2481BF0A348497113F8A5B51AFCAF6C1BF86E5FA4 6C1BF0A26E5F6C1BE0A36C6D4D13C0A26C6D4D1380A26C1B006C6D4D5A6E5E6C626D6C4C 5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B90C7FC6D6D4B5A6D01FF02035B023F01 E0011F13F0020F01FC90B512C0020390B7C8FC020016FC031F15E0030392C9FCDB001F13 E0565479D265>79 DI82 D<003FBC1280A59126C0003F9038C0007F49C71607D87FF8060113C001E08449197F4919 3F90C8171FA2007E1A0FA3007C1A07A500FC1BE0481A03A6C994C7FCB3B3AC91B912F0A5 53517BD05E>84 D87 D97 DI<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A1FFE0001 FFD97FFC491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F1300705A48 92C8FC5BA312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F806C6DEC 3F006C6D147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A01001580023F49C7FC 020113E033387CB63C>I<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91 B6FC010315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980 485B83485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C 6D49B5FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9 003F13F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC010715C04901 817F903A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191 C7FC4817E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01 F06E14037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE 903A0FFFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>IIII<137F497E 000313E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512017EB3B3 A6B612E0A51B547BD325>I108 DII<913801FFE0021F13FE91B6 12C0010315F0010F9038807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F4849 6D7F4A147F48834890C86C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A3 003F18006D5DA26C5FA26C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE 011F90C7FC903A0FFF807FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I< 903A3FF001FFE0B5010F13FE033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9 FFE0EB1FFFC602806D7F92C76C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A 0060A36118FFA2615F616E4A5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FB FE075B02F8B612E06F1480031F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I< 90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F83FFC0003D9E3E013 FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90C7FCA55CB3AAB612 FCA52F367CB537>114 D<903903FFF00F013FEBFE1F90B7FC120348EB003FD80FF81307 D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C13FF15 F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714E0EC003F0307 13F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6DEC3F8001F8EC 7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C387CB6 35>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6FCB8FC A426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0FC6DEB FFF86D6C5B021F5B020313802A4D7ECB34>II<007FB5 00F090387FFFFEA5C66C48C7000F90C7FC6D6CEC07F86D6D5C6D6D495A6D4B5A6F495A6D 6D91C8FC6D6D137E6D6D5B91387FFE014C5A6E6C485A6EEB8FE06EEBCFC06EEBFF806E91 C9FCA26E5B6E5B6F7E6F7EA26F7F834B7F4B7F92B5FCDA01FD7F03F87F4A486C7E4A486C 7E020F7FDA1FC0804A486C7F4A486C7F02FE6D7F4A6D7F495A49486D7F01076F7E49486E 7E49486E7FEBFFF0B500FE49B612C0A542357EB447>120 D I<001FB8FC1880A3912680007F130001FCC7B5FC01F0495B495D49495B495B4B5B48C75C 5D4B5B5F003E4A90C7FC92B5FC4A5B5E4A5B5CC7485B5E4A5B5C4A5B93C8FC91B5FC495B 5D4949EB0F805B495B5D495B49151F4949140092C7FC495A485E485B5C485E485B4A5C48 495B4815074849495A91C712FFB8FCA37E31357CB43C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmbx10 10.95 45 /Fl 45 121 df12 D40 D<127012F8127C7EEA3F806C7E6C7E12076C7E7F6C7E6C7EA2137F8013 3F806D7EA280130FA280130780A36D7EA4807FA51580B01500A55B5CA4495AA35C130F5C A2131F5CA2495A5C137F91C7FC13FEA2485A485A5B485A120F485A485A003EC8FC5A5A12 70195A7AC329>I44 D46 D48 D<140F143F5C495A130F48B5FCB6FCA313F7EAFE071200B3B3A8B712F0A5243C78BB34> I<903803FF80013F13F890B512FE00036E7E4881260FF80F7F261FC0037F4848C67F486C 6D7E6D6D7E487E6D6D7EA26F1380A46C5A6C5A6C5A0007C7FCC8FC4B1300A25E153F5E4B 5AA24B5A5E4A5B4A5B4A48C7FC5D4A5AEC1FE04A5A4A5A9139FF000F80EB01FC495A4948 EB1F00495AEB1F8049C7FC017E5C5B48B7FC485D5A5A5A5A5AB7FC5EA4293C7BBB34>I< 903801FFE0010F13FE013F6D7E90B612E04801817F3A03FC007FF8D807F06D7E82D80FFC 131F6D80121F7FA56C5A5E6C48133FD801F05CC8FC4B5A5E4B5A4A5B020F5B902607FFFE C7FC15F815FEEDFFC0D9000113F06E6C7E6F7E6F7E6F7E1780A26F13C0A217E0EA0FC048 7E487E487E487EA317C0A25D491580127F49491300D83FC0495A6C6C495A3A0FFE01FFF8 6CB65A6C5DC61580013F49C7FC010313E02B3D7CBB34>II<00071538D80FE0EB01F801FE133F90B6FC5E5E5E5E93 C7FC5D15F85D15C04AC8FC0180C9FCA9ECFFC0018713FC019F13FF90B67E020113E09039 F8007FF0496D7E01C06D7E5B6CC77FC8120F82A31780A21207EA1FC0487E487E12FF7FA2 1700A25B4B5A6C5A01805C6CC7123F6D495AD81FE0495A260FFC075B6CB65A6C92C7FCC6 14FC013F13F0010790C8FC293D7BBB34>I<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA2 4B7FA34B7F157C03FC7FEDF87FA2020180EDF03F0203804B7E02078115C082020F814B7E 021F811500824A81023E7F027E81027C7FA202FC814A147F49B77EA34982A2D907E0C700 1F7F4A80010F835C83011F8391C87E4983133E83017E83017C81B500FC91B612FCA5463F 7CBE4F>65 D<922607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE01 039138803FFB499039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F484915 07485B4A1503481701485B18004890CAFC197E5A5B193E127FA349170012FFAC127F7F19 3EA2123FA27F6C187E197C6C7F19FC6C6D16F86C6D150119F06C6D15036C6DED07E0D97F FEED0FC06D6CED3F80010F01C0ECFF006D01F8EB03FE6D9039FF801FFC010091B55A023F 15E002071580020002FCC7FC030713C03F407ABE4C>67 DII77 D80 DII<903A03FFC001C0011FEBF803017FEBFE0748B6128F4815DF48010013FFD80FF8130F 48481303497F4848EB007F127F49143F161F12FF160FA27F1607A27F7F01FC91C7FCEBFF 806C13F8ECFFC06C14FCEDFF806C15E016F86C816C816C816C16806C6C15C07F010715E0 EB007F020714F0EC003F1503030013F8167F163F127800F8151FA2160FA27EA217F07E16 1F6C16E06D143F01E015C001F8EC7F8001FEEB01FF9026FFE00713004890B55A486C14F8 D8F81F5CD8F00314C027E0003FFEC7FC2D407ABE3A>I<003FB912FCA5903BFE003FFE00 3FD87FF0EE0FFE01C0160349160190C71500197E127EA2007C183EA400FC183F48181FA5 C81600B3AF010FB712F8A5403D7CBC49>I91 D93 D<903807FFC0013F13F848 B6FC48812607FE037F260FF8007F6DEB3FF0486C806F7EA36F7EA26C5A6C5AEA01E0C8FC 153F91B5FC130F137F3901FFFE0F4813E0000F1380381FFE00485A5B485A12FF5BA4151F 7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFFE100031480C6EC003FD91F F890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE0020713FC021FEBFF80027F80 DAFF8113F09139FC003FF802F06D7E4A6D7E4A13074A80701380A218C082A318E0AA18C0 A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF01FFE0496CB55AD9F01F91 C7FCD9E00713FCC7000113C033407DBE3A>II< EE07F8ED07FFA5ED003F161FAFEC7FF0903807FFFE011FEBFF9F017F14DF9039FFF01FFF 48EBC00348EB00014848EB007F485A001F153F5B123FA2127F5BA212FFAA127FA37F123F A26C6C147F120F6D14FF6C6C01037F6C6D48EBFFE06CEBF03F6C6CB512BF6D143F010713 FC010001E0EBE00033407DBE3A>III<903A03FF8007F0013F9038F83FF8499038FCFFFC48B712FE48018313F93A07FC 007FC34848EB3FE1001FEDF1FC4990381FF0F81700003F81A7001F5DA26D133F000F5D6C 6C495A3A03FF83FF8091B5C7FC4814FC01BF5BD80F03138090CAFCA2487EA27F13F06CB6 FC16F016FC6C15FF17806C16C06C16E01207001F16F0393FE000034848EB003F49EC1FF8 00FF150F90C81207A56C6CEC0FF06D141F003F16E001F0147FD81FFC903801FFC02707FF 800F13006C90B55AC615F8013F14E0010101FCC7FC2F3D7DA834>I<13FFB5FCA512077E AFED1FF8EDFFFE02036D7E4A80DA0FE07F91381F007F023C805C4A6D7E5CA25CA35CB3A4 B5D8FE0FB512E0A5333F7CBE3A>III<13FFB5FCA512077EB092380FFFFEA5DB01FEC7FC4B5A ED07F0ED1FE04B5A4B5A4BC8FCEC03FC4A5A4A5A141FEC7FF84A7EA2818102E77F02C37F 148102007F826F7E6F7E151F6F7E826F7F6F7F816F7FB5D8FC07EBFFC0A5323F7DBE37> I<13FFB5FCA512077EB3B3AFB512FCA5163F7CBE1D>I<01FFD91FF8ECFFC0B590B50107 13F80203DAC01F13FE4A6E487FDA0FE09026F07F077F91261F003FEBF8010007013EDAF9 F0806C0178ECFBC04A6DB4486C7FA24A92C7FC4A5CA34A5CB3A4B5D8FE07B5D8F03FEBFF 80A551297CA858>I<01FFEB1FF8B5EBFFFE02036D7E4A80DA0FE07F91381F007F000701 3C806C5B4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A533297CA83A>II<01FFEBFFE0B5000713FC021FEBFF80027F80DAFF81 13F09139FC007FF8000701F06D7E6C496D7E4A130F4A6D7E1880A27013C0A38218E0AA4C 13C0A318805E18005E6E5C6E495A6E495A02FCEBFFF0DAFF035B92B55A029F91C7FC0287 13FC028113C00280C9FCACB512FEA5333B7DA83A>I<3901FE01FE00FF903807FF804A13 E04A13F0EC3F1F91387C3FF8000713F8000313F0EBFFE0A29138C01FF0ED0FE091388007 C092C7FCA391C8FCB3A2B6FCA525297DA82B>114 D<90383FFC1E48B512BE000714FE5A 381FF00F383F800148C7FC007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C13FF15 C06C14F06C806C806C806C80C61580131F1300020713C014000078147F00F8143F151F7E A27E16806C143F6D140001E013FF9038F803FE90B55A15F0D8F87F13C026E00FFEC7FC22 2B7DA929>III 119 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr8 8 16 /Fm 16 117 df22 D<13031307130E131C1338137013F0EA01E013 C01203EA0780A2EA0F00A2121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA07 80A2EA03C0120113E0EA00F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2EA00F0A21378A3133CA4131EA513 1FA2130FAB131FA2131EA5133CA41378A313F0A2EA01E0A2EA03C013801207EA0F00120E 5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387F FFFEA2172C7AAB23>III<140EA2141E143EA2 147E14FEA2EB01BE1303143E1306130E130C131813381330136013E013C0EA0180120313 001206120E120C5A123812305A12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC 23>I61 D<15F8141FA214011400ACEB0FE0EB 7FF83801F81E3803E0073807C003380F8001EA1F00481300123E127EA25AA9127C127EA2 003E13017EEB8003000F13073903E00EFC3A01F03CFFC038007FF090391FC0F800222F7E AD27>100 DI<013F13F89038FFC3FE3903E1FF 1E3807807C000F140C391F003E00A2003E7FA76C133EA26C6C5A00071378380FE1F0380C FFC0D81C3FC7FC90C8FCA3121E121F380FFFF814FF6C14C04814F0391E0007F848130048 147C12F848143CA46C147C007C14F86CEB01F06CEB03E03907E01F803901FFFE0038003F F01F2D7E9D23>103 D 108 D111 D<1360A413E0A312011203A21207121FB5 12F0A23803E000AF1418A714383801F03014703800F860EB3FE0EB0F80152A7FA81B> 116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr17 17.28 26 /Fn 26 122 df45 D<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3F C0EA0F000C0C748B24>I<170FA34D7EA24D7EA34D7EA34D7EA34C7F17DFA29338039FFC 178FA29338070FFE1707040F7FEE0E03A2041E80EE1C01A2043C80EE3800A24C80187FA2 4C80183FA24B4880181F0303814C130FA203078193C71207A24B81030E80A24B8284A24B 8284A24B82197F03F0824B153FA20201834B151FA202038392B8FCA24A83A292C9120702 0E8385A24A8485023C84023882A20278840270177FA202F0844A173FA24948841A1FA249 48841A0FA249CB7F1A074985865B496C85497E48486C4D7F000F01F8051F13F0B60407B6 12F0A45C657DE463>65 D77 DI80 D83 D85 D97 D<4AB47E020F13F8023F13FE9139FF007F80D903FCEB07 E0D907F0EB01F0D91FE0EB007849488049488049C87E48485D4915FF00034B138048485C A2485AA2485AA2003F6F130049EC007C94C7FC127FA35B12FFAD127F7FA4123F7FA2001F EE01C07F000F16036D168012076C6C15076D160000015E6C6C151E6D6C5C6D6C5C6D6C5C D90FF8495AD903FCEB07C0903A00FF803F8091263FFFFEC7FC020F13F80201138032417C BF3A>99 D<181EEF3FFEEE07FFA4EE000F1703A21701B3AAEDFF80020F13F8023F13FE91 39FF803F81903A03FC0007C14948EB01E1D91FE0EB00F94948147D4948143D49C8121F48 48150F491507120348481503491501120F121F5BA2123F5B127FA45B12FFAD127F7FA312 3FA27F121FA26C6C1503A26C6C150712036D150F6C6C151F0000163D137F6D6CECF9FF6D 6CEB01F1D90FF0D903C113C06D6CD90F81EBFF80D901FFEB7F019039007FFFFC021F13E0 0201010091C7FC41657CE349>II103 DI<133C13FF487F 487FA66C5B6C90C7FC133C90C8FCB3A2EB03C0EA07FF127FA41201EA007FA2133FB3B3AC 497E497EB612E0A41B5F7DDE23>I108 DIIII<9039078003F8D807FFEB0FFFB5013F13C092387C0FE0913881 F01F9238E03FF00001EB838039007F8700148FEB3F8E029CEB1FE0EE0FC00298EB030002 B890C7FCA214B014F0A25CA55CB3B0497EEBFFF8B612FCA42C3F7CBE33>114 D<9139FFE00180010FEBFC03017FEBFF073A01FF001FCFD803F8EB03EFD807E0EB01FF48 487F4848147F48C8123F003E151F007E150F127CA200FC1507A316037EA27E7F6C7E6D91 C7FC13F8EA3FFE381FFFF06CEBFF806C14F86C14FF6C15C06C6C14F0011F80010714FED9 007F7F02031480DA003F13C01503030013E0167F00E0ED1FF0160F17F86C15071603A36C 1501A37EA26C16F016037E17E06D14076DEC0FC06D1580D8FDF0141FD8F8F8EC7F00013E 14FC3AF01FC00FF80107B512E0D8E001148027C0003FF8C7FC2D417DBF34>I<1438A714 78A414F8A31301A31303A21307130F131FA2137F13FF1203000F90B6FCB8FCA3260007F8 C8FCB3AE17E0AE6D6CEB01C0A316036D6C148016076D6C14006E6C5A91383FC01E91381F F07C6EB45A020313E09138007F802B597FD733>III 121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsy10 10.95 25 /Fo 25 112 df<007FB812FEBAFCA26C17FE3804799847>0 D<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A799B19>I<0060166000F816F06C1501007E15036CED07E06C6CEC 0FC06C6CEC1F806C6CEC3F006C6C147E6C6C5C6C6C495A017E495A6D495A6D6C485A6D6C 485A6D6C48C7FC903803F07E6D6C5A903800FDF8EC7FF06E5A6E5AA24A7E4A7EECFDF890 3801F8FC903803F07E49487E49486C7E49486C7E49486C7E017E6D7E496D7E48486D7E48 48147E4848804848EC1F804848EC0FC048C8EA07E0007EED03F048150148150000601660 2C2C73AC47>I15 D<0203B612FE023F15FF91B8FC010316FED90FFEC9FCEB1FE0EB7F8001FECAFCEA01F848 5A485A485A5B48CBFCA2123EA25AA21278A212F8A25AA87EA21278A2127CA27EA27EA26C 7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FE0EB0FFE0103B712FE010016FF143F020315FE91 CAFCAE001FB812FE4817FFA26C17FE384879B947>18 D<180E183F18FFEF03FEEF0FF8EF 3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED7FE0913801FF80DA 07FEC8FCEC1FF8EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC0 48CBFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1F F0EC07FE913801FF809138007FE0ED1FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE9338 00FF80EF3FE0EF0FF8EF03FEEF00FF183F180E1800AE007FB812FEBAFCA26C17FE384879 B947>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF 9038007FC0EC1FF0EC07FE913801FF809138007FE0ED1FF8ED03FE923800FF80EE3FE0EE 0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FFA2EF03FEEF0FF8EF3FE0EFFF8093 3803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1F F8EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC12 70CCFCAE007FB812FEBAFCA26C17FE384879B947>I<0203B612FE023F15FF91B8FC0103 16FED90FFEC9FCEB1FE0EB7F8001FECAFCEA01F8485A485A485A5B48CBFCA2123EA25AA2 1278A212F8A25AA87EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB 1FE0EB0FFE0103B712FE010016FF143F020315FE383679B147>26 D<19301978A2197C193CA2193E191EA2191F737EA2737E737EA2737E737E1A7C1A7EF21F 80F20FC0F207F0007FBB12FCBDFCA26C1AFCCDEA07F0F20FC0F21F80F27E001A7C624F5A 4F5AA24F5A4F5AA24FC7FC191EA2193E193CA2197C1978A2193050307BAE5B>33 D<0203B512F8023F14FC91B6FC010315F8D90FFEC8FCEB1FE0EB7F8001FEC9FCEA01F848 5A485A485A5B48CAFCA2123EA25AA21278A212F8A25AA2B812F817FCA217F800F0CAFCA2 7EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FE0EB0FFE0103B6 12F8010015FC143F020314F82E3679B13D>50 D<1718173C177CA217F8A2EE01F0A2EE03 E0A2EE07C0160F1780EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2 153E157E157C5DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25C13015C495AA2 495AA2495AA249C9FCA2133EA25BA25BA2485AA2485AA2485A120F5B48CAFCA2123EA25A A25AA25A12602E5474C000>54 D<126012F0AE12FCA412F0AE126006227BA700>I<0060 EE018000F0EE03C06C1607A200781780007C160FA2003C1700003E5EA26C163EA26C163C 6D157CA2000716786D15F8A26C6C4A5AA200015E6D140390B7FC6C5EA3017CC7EA0F80A2 013C92C7FC013E5CA2011E141E011F143EA26D6C5BA2010714786E13F8A26D6C485AA201 015CECF003A201005CECF807A291387C0F80A2023C90C8FCEC3E1FA2EC1E1EEC1F3EA2EC 0FFCA26E5AA36E5AA36E5A6E5A324180BE33>I67 D<047FB612FC0307B8FC031F1780157F4AB9FC912903F80FE0 00011300DA0FC0ED007EDA1F00167C023E17604A011F92C7FC02FC5C495AA213034A495A 495A5C0106C7FC90C848CAFCA3167E16FEA34B5AA35E150393B612F0A24B5D614B92C8FC 04E0CAFC5E151F5EA24BCBFCA25D157E15FE5DA24A5AA24A5AA24A5AA20003495A121F48 6C485A127F486C48CCFCEBE03E387FFC7CEBFFF86C5B6C13C06C5BD801FCCDFC49417FBD 41>70 D<033FB612F00207B7FC023F16E091B812800103EEFE0090280FFC0007C0C7FCD9 1F80130F013EC7485A4992C8FC01FC5C48485C167E484814FE01C05C90C8FCC812015E15 03A34B5AA35E150FA34B5AA44B5AA44BC9FCA415FEA35D1401A25D14035DA24A5A18704A 48EB01F04D5A4A48130792C7485A023E5D4A023FC7FC0007B712FE001F16F8485E481680 B700FCC8FC3C3E83BD32>73 D86 DI92 D<15C04A7E4A7EA24A7EA34A7EA2EC1F3EA2EC3E1FA2EC3C0F027C7FA24A6C7EA249486C 7EA2ECE001010380A249486C7EA24948137CA249C77EA2011E141E013E141FA2496E7EA2 496E7EA2491403000182A248486E7EA248486E7EA2491578000F167CA248C97EA2003E82 A2003C82007C1780A248EE07C0A24816030060EE018032397BB63D>94 D<153FEC03FFEC0FE0EC3F80EC7E00495A5C495AA2495AB3AA130F5C131F495A91C7FC13 FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE133F806D7E130F801307B3AA6D7EA26D7E 80EB007EEC3F80EC0FE0EC03FFEC003F205B7AC32D>102 D<12FCEAFFC0EA07F0EA01FC EA007E6D7E131F6D7EA26D7EB3AA801303806D7E1300147FEC1FC0EC07FEEC00FFEC07FE EC1FC0EC7F0014FC1301495A5C13075CB3AA495AA2495A133F017EC7FC485AEA07F0EAFF C000FCC8FC205B7AC32D>I<126012F0B3B3B3B3B11260045B76C319>106 D<126012F07EA21278127CA2123C123EA2121E121FA27E7FA212077FA212037FA212017F A212007FA21378137CA27FA2131E131FA27F80A2130780A2130380A2130180A2130080A2 1478147CA2143C143EA2141E141FA26E7EA2140781A2140381A2140181A2140081A21578 157CA2153C153EA2151E151FA2811680A2150716C0A21503ED0180225B7BC32D>110 D<127CB4FCEA7FC0EA0FE01207EA03F0A2EA01F8A3EA00FCAAEA01F8A4EA03F0A2EA07E0 A3EA0FC0A2EA1F80A3EA3F00A2127EA45AAA127EA37EA2EA1F8013C0EA0FF8EA03FCEA00 F80E3C7BB419>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmtt10 10.95 15 /Fp 15 120 df<120FEA3FC0EA7FE0A2EAFFF0A4EA7FE0A2EA3FC0EA0F000C0C6E8B30> 46 D64 D97 D99 D<49B4FC010713E0011F13F8017F7F 90B57E488048018113803A07FC007FC04848133FD81FE0EB1FE0150F484814F049130712 7F90C7FCED03F85A5AB7FCA516F048C9FC7E7EA27F003FEC01F06DEB03F86C7E6C7E6D13 07D807FEEB1FF03A03FFC07FE06C90B5FC6C15C0013F14806DEBFE00010713F8010013C0 252A7CA830>101 DI104 D106 D<02FC137E3B7FC3FF01FF80D8FF EF01877F90B500CF7F15DF92B57E6C010F13872607FE07EB03F801FC13FE9039F803FC01 A201F013F8A301E013F0B3A23C7FFE0FFF07FF80B548018F13C0A46C486C010713803228 81A730>109 DI<49B4FC010F13E0013F13F8497F90 B57E0003ECFF8014013A07FC007FC04848EB3FE0D81FE0EB0FF0A24848EB07F849130300 7F15FC90C71201A300FEEC00FEA86C14016C15FCA26D1303003F15F86D13076D130F6C6C EB1FF06C6CEB3FE06D137F3A07FF01FFC06C90B512806C15006C6C13FC6D5B010F13E001 0190C7FC272A7CA830>I114 D<90381FFC1E48B5129F000714FF5A5A5A38 7FF007EB800100FEC7FC4880A46C143E007F91C7FC13E06CB4FC6C13FC6CEBFF806C14E0 000114F86C6C7F01037F9038000FFF02001380007C147F00FEEC1FC0A2150F7EA27F151F 6DEB3F806D137F9039FC03FF0090B6FC5D5D00FC14F0D8F83F13C026780FFEC7FC222A79 A830>II<3B3FFFC01FFFE0486D4813F0B515F8A26C16F06C496C13E0D807E0 C7EA3F00A26D5C0003157EA56D14FE00015DEC0F80EC1FC0EC3FE0A33A00FC7FF1F8A214 7DA2ECFDF9017C5C14F8A3017E13FBA290393FF07FE0A3ECE03FA2011F5C90390F800F80 2D277FA630>119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmss10 10.95 11 /Fq 11 111 df70 D76 D78 D<4AB47E020F13F0027F13FE91B6FC01 0315C04981011F010013F8D93FF8EB1FFCD97FE0EB07FE4A130349486D7E4890C8138048 48ED7FC049153F4848ED1FE04848ED0FF0A24848ED07F8A2491503003F17FCA249150100 7F17FEA390CAFC4817FFAC6D5D007F17FEA46D1503003F17FCA26D1507001F17F86D150F 000F17F06D151F6C6CED3FE0A26C6CED7FC06C6CEDFF806C6D4913006E5BD97FF0EB0FFE 6D6C495A6DB4EBFFF8010790B512E06D5D010092C7FC6E5B020F13F00201138038437BC0 43>I82 D84 D87 D97 D<49B47E010F13F0013F13FC4913FF90B612805A48 1300D807FCEB1F00D80FF0130748487F4990C7FC123F5B127F90C9FCA312FEAA127FA36C 7EA26C6C14406DEB01C06C6C13036C6C131F01FF13FF6C90B5FC7E6C6C14806DEBFE0001 0F13F001011380222B7DA928>99 D101 D<38FC01FF010713C0011F13F0 017F13F890B512FC12FD39FFF80FFEEBE003EBC00190388000FFA290C7127FA35AB3A920 2979A82F>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 10.95 90 /Fr 90 128 df<16E04B7EA24B7EA24B7EA24B7EA2ED1DFFA203387FA29238787FC01570 9238F03FE015E002016D7E15C002036D7E158002076D7E15004A6D7E140E021E6D7E141C 023C6D7F143802786E7E147002F06E7E5C01016F7E5C01036F7E5C01076F7E91C8FC496F 7E130E011E6F7E131C013C6F7F13380178707E137001F0707E5B0001717E5B0003717E5B 0007717E90CAFC48717E120E001E717E001FBAFC481980A24819C0A2BB12E0A243417CC0 4C>1 DI5 D<4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383F F03FD907F090397FE07F80494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01 FE6D91C7FC82ADB97EA3C648C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35 >11 D<4AB4FC021F13C091387F01F0903901FC0078D907F0131C4948133E494813FF4948 5A137F1400A213FE6F5A163893C7FCAA167FB8FCA33900FE00018182B3AC486CECFF8000 7FD9FC3F13FEA32F407FBF33>I<4AB47E021F13F791387F00FFEB01F8903807F001EB0F E0EB1FC0EB3F80137F14008101FE80AEB8FCA3C648C77EB3AE486CECFF80007FD9FC3F13 FEA32F407FBF33>I<4AB4ECFF80021FD9C00F13E0913B7F01F03F80F8903C01F80078FE 003CD907F0D93FF8130E49484948131F49484948EB7F804948484913FF137F02005CA201 FE92C7FC6FED7F0070141C96C7FCAAF13F80BBFCA3C648C76CC7FC197F193FB3AC486C4A 6CEB7FC0007FD9FC3FD9FE1FB5FCA348407FBF4C>I16 D<133E133F137F13FFA2EA01FEEA03FCEA07F813F0EA0F E0EA1FC01380EA3E005A5A1270122010116EBE2D>19 D22 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AC121EAB120CC7FCA8121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A4179C019>33 D<001E130F397F803FC000FF137F01C013E0A201E013 F0A3007F133F391E600F3000001300A401E01370491360A3000114E04913C00003130101 001380481303000EEB070048130E0018130C0038131C003013181C1C7DBE2D>I<14E0A4 EB07FC90383FFF8090B512E03901F8E3F03903E0E0FCD807C0133CD80F807FD81F007F00 3E80003C1580007C140316C00078141F00F8143F157FA47EED3F806CEC0E0092C7FC127F 138013C0EA3FF013FEEA1FFF6C13FC6C13FF6C14C06C806C6C13F8011F7F130301007FEC E7FF14E102E01380157F153FED1FC0A2003E140F127FD8FF801307A5130000FC158000F0 140F1270007815005D6C141E153E6C5C6C5C3907C0E1F03903F8EFE0C6B51280D93FFEC7 FCEB0FF8EB00E0A422497BC32D>36 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200 A413E013C0A312011380120313005A120E5A1218123812300B1C79BE19>39 D<1430147014E0EB01C0EB03801307EB0F00131E133E133C5B13F85B12015B1203A2485A A2120F5BA2121F90C7FCA25AA3123E127EA6127C12FCB2127C127EA6123E123FA37EA27F 120FA27F1207A26C7EA212017F12007F13787F133E131E7FEB07801303EB01C0EB00E014 701430145A77C323>I<12C07E12707E7E121E7E6C7E7F12036C7E7F12007F1378137CA2 7FA2133F7FA21480130FA214C0A3130714E0A6130314F0B214E01307A614C0130FA31480 A2131F1400A25B133EA25BA2137813F85B12015B485A12075B48C7FC121E121C5A5A5A5A 145A7BC323>I<1506150FB3A9007FB912E0BA12F0A26C18E0C8000FC9FCB3A915063C3C 7BB447>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380 120313005A120E5A1218123812300B1C798919>II<121EEA7F80 A2EAFFC0A4EA7F80A2EA1E000A0A798919>II IIII<150E15 1E153EA2157EA215FE1401A21403EC077E1406140E141CA214381470A214E0EB01C0A2EB 0380EB0700A2130E5BA25B5BA25B5B1201485A90C7FC5A120E120C121C5AA25A5AB8FCA3 C8EAFE00AC4A7E49B6FCA3283E7EBD2D>I<00061403D80780131F01F813FE90B5FC5D5D 5D15C092C7FC14FCEB3FE090C9FCACEB01FE90380FFF8090383E03E090387001F8496C7E 49137E497F90C713800006141FC813C0A216E0150FA316F0A3120C127F7F12FFA416E090 C7121F12FC007015C012780038EC3F80123C6CEC7F00001F14FE6C6C485A6C6C485A3903 F80FE0C6B55A013F90C7FCEB07F8243F7CBC2D>II<1238123C123F90B612FCA316F85A16F016E00078C712010070EC03C0ED 078016005D48141E151C153C5DC8127015F04A5A5D14034A5A92C7FC5C141EA25CA2147C 147814F8A213015C1303A31307A3130F5CA2131FA6133FAA6D5A0107C8FC26407BBD2D> III<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A2779A619>I<121EEA7F80A2EAFFC0A4EA7F80A2 EA1E00C7FCB3121E127FEAFF80A213C0A4127F121E1200A412011380A3120313005A1206 120E120C121C5A1230A20A3979A619>I<007FB912E0BA12F0A26C18E0CDFCAE007FB912 E0BA12F0A26C18E03C167BA147>61 D63 D<15074B7EA34B7EA34B7EA34B7EA34B7E 15E7A2913801C7FC15C3A291380381FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D 7EA34A6D7EA349486D7E91B6FCA249819138800001A249C87EA24982010E157FA2011E82 011C153FA2013C820138151FA2017882170F13FC00034C7ED80FFF4B7EB500F0010FB512 F8A33D417DC044>65 DIIIIIIII<011FB512FCA3D9000713006E 5A1401B3B3A6123FEA7F80EAFFC0A44A5A1380D87F005B007C130700385C003C495A6C49 5A6C495A2603E07EC7FC3800FFF8EB3FC026407CBD2F>IIII< B56C91B512F88080D8007F030713006EEC01FC6E6E5A1870EB77FCEB73FEA2EB71FF0170 7FA26E7E6E7EA26E7E6E7EA26E7E6E7EA26E7E6E7FA26F7E6F7EA26F7E6F7EA26F7E6F7E A26F7E6F1380A2EE7FC0EE3FE0A2EE1FF0EE0FF8A2EE07FCEE03FEA2EE01FF7013F0A217 7F173FA2171F170FA2170701F81503487ED807FF1501B500F81400A218703D3E7DBD44> III< B712C016FCEEFF800001D9C00013E06C6C48EB1FF0EE07FCEE01FE707E84717EA2717EA2 84A760177F606017FF95C7FCEE01FCEE07F8EE1FE0EEFF8091B500FCC8FC16F091388001 FCED003FEE1FC0707E707E83160383160183A383A484A4F0C004190EA28218E0057F131E 2601FFE0161CB600C0EB3FF094381FF83805071370CA3801FFE09438003F803F407DBD43 >82 DI<003FB91280A3903AF0 007FE001018090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00 E0A5C81600B3B14B7E4B7E0107B612FEA33B3D7DBC42>IIII<003F B712F8A391C7EA1FF013F801E0EC3FE00180EC7FC090C8FC003EEDFF80A2003C4A130000 7C4A5A12784B5A4B5AA200704A5AA24B5A4B5AA2C8485A4A90C7FCA24A5A4A5AA24A5AA2 4A5A4A5AA24A5A4A5AA24990C8FCA2495A4948141CA2495A495AA2495A495A173C495AA2 4890C8FC485A1778485A484815F8A24848140116034848140F4848143FED01FFB8FCA32E 3E7BBD38>90 DI<486C13C00003 130101001380481303000EEB070048130E0018130C0038131C0030131800701338006013 30A300E01370481360A400CFEB678039FFC07FE001E013F0A3007F133FA2003F131F01C0 13E0390F0007801C1C73BE2D>II< EB0FF8EBFFFE3903F01F8039078007E0000F6D7E9038E001F8D81FF07F6E7EA3157F6C5A EA0380C8FCA4EC1FFF0103B5FC90381FF87FEB7F803801FC00EA07F8EA0FE0485A485AA2 48C7FCEE038012FEA315FFA3007F5BEC03BF3B3F80071F8700261FC00E13CF3A07F03C0F FE3A01FFF807FC3A003FC001F0292A7DA82D>97 DI< 49B4FC010F13E090383F00F8017C131E4848131F4848137F0007ECFF80485A5B121FA248 48EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3003FEC01C07F001F140316806C6C1307 6C6C14000003140E6C6C131E6C6C137890383F01F090380FFFC0D901FEC7FC222A7DA828 >IIII<167C903903F801FF903A1FFF078F8090397E0FDE1F9038 F803F83803F001A23B07E000FC0600000F6EC7FC49137E001F147FA8000F147E6D13FE00 075C6C6C485AA23901F803E03903FE0FC026071FFFC8FCEB03F80006CAFC120EA3120FA2 7F7F6CB512E015FE6C6E7E6C15E06C810003813A0FC0001FFC48C7EA01FE003E14004815 7E825A82A46C5D007C153E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017F C7FC90383FFFFC010313C0293D7EA82D>III<1478EB01FEA2EB 03FFA4EB01FEA2EB00781400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14 FEA214FCEB81F8EA7F01387C03F0381E07C0380FFF803801FC00185185BD1C>III<2701F801FE14FF00FF902707FFC0 0313E0913B1E07E00F03F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F 000101805C01FBD900FF147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F8 7FD9FC3F13FEA347287DA74C>I<3901F801FE00FF903807FFC091381E07E091387803F0 00079038E001F82603F9C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8 F87F13FCA32E287DA733>I<14FF010713E090381F81F890387E007E01F8131F4848EB0F 804848EB07C04848EB03E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C 15FEA36C6CEB01FCA3001F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D800 7EEB7E0090383F81FC90380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381F FF8091387C0FE09039FDE003F03A07FFC001FC6C496C7E6C90C7127F49EC3F805BEE1FC0 17E0A2EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001 F89039FDE007F09039FC780FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733 >I<02FF131C0107EBC03C90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC 150748481303121F485A1501485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C13 0F6C7E6C6C133DD8007E137990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512 F8A32D3A7DA730>I<3901F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB 1401EC00FC01FF1330491300A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901 FFF8E03807C03F381F000F003E1307003C1303127C0078130112F81400A27E7E7E6D1300 EA7FF8EBFFC06C13F86C13FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF8 00E01303A214017E1400A27E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0 FFFC38C01FE01D2A7DA824>I<131CA6133CA4137CA213FCA2120112031207001FB512C0 B6FCA2D801FCC7FCB3A215E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FE EB01F81B397EB723>IIIIII<001FB61280 A2EBE0000180140049485A001E495A121C4A5A003C495A141F00385C4A5A147F5D4AC7FC C6485AA2495A495A130F5C495A90393FC00380A2EB7F80EBFF005A5B4848130712074914 00485A48485BA248485B4848137F00FF495A90B6FCA221277EA628>III<001C130E007FEB3F8039FF807FC0A5397F003F80001C EB0E001A0977BD2D>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr12 14.4 15 /Fs 15 122 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: Letter letter %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 911 91 a Fs(The)39 b(Computational)33 b(Complexit)m(y)i(Column) 1902 317 y Fr(b)m(y)1599 543 y Fq(Lance)c(F)m(ORTNO)m(W)1475 769 y Fr(NEC)f(Researc)m(h)h(Institute)986 882 y(4)g(Indep)s(endence)e (W)-8 b(a)m(y)g(,)33 b(Princeton,)d(NJ)g(08540,)j(USA)1306 995 y Fp(fortnow@research.nj.nec)o(.com)834 1108 y Fr(h)m (ttp://www.neci.nj.nec.com/homepages/fortno)m(w/b)s(eatcs)141 1412 y(This)c(has)h(b)s(een)f(an)i(exciting)f(summer)f(for)h (computational)g(complexit)m(y)-8 b(.)136 1595 y Fo(\017)46 b Fr(Manindra)21 b(Agra)m(w)m(al)i(and)f(his)f(studen)m(ts)h(Neera)5 b(j)23 b(Ka)m(y)m(al)g(and)f(Nitin)f(Saxena)h(at)h(ITT)f(Kanpur)e(ha)m (v)m(e)k(giv)m(en)227 1708 y(the)j(\014rst)f(pro)m(v)-5 b(ably)26 b(deterministic)f(p)s(olynomial-time)f(algorithm)i(for)g (primalit)m(y)-8 b(.)38 b(F)-8 b(or)28 b(more)f(details)e(see)227 1821 y(Josep)31 b(D)-10 b(\023)-35 b(\020az's)30 b(Algorithmic)f (Column)g(in)g(this)g(bulletin.)136 2007 y Fo(\017)46 b Fr(Madh)m(u)25 b(Sudan)e(receiv)m(ed)i(the)g(Nev)-5 b(anlinna)24 b(Prize)g(for)g(his)g(w)m(ork)h(on)f(probabilistically)d (c)m(hec)m(k)-5 b(able)26 b(pro)s(ofs)227 2120 y(and)e (error-correcting)h(co)s(des.)39 b(This)23 b(prize)g(is)h(giv)m(en)g (with)g(the)g(Fields)f(medal)h(ev)m(ery)i(four)d(y)m(ears)j(for)e(w)m (ork)227 2233 y(in)29 b(\\information)g(science".)136 2419 y Fo(\017)46 b Fr(The)40 b(Conference)g(on)g(Computational)f (Complexit)m(y)h(held)e(in)h(Mon)m(treal)i(in)e(Ma)m(y)j(brok)m(e)e (all)f(records)227 2532 y(with)h(140)i(participan)m(ts.)70 b(Next)42 b(y)m(ear's)f(conference)h(is)e(coming)g(to)i(Europ)s(e)d (and)h(will)e(b)s(e)i(held)g(July)227 2645 y(7-10)33 b(in)d(Aarh)m(us,)i(Denmark.)44 b(See)31 b(the)h(call)f(for)g(pap)s (ers)f(elsewhere)h(in)f(this)g(bulletin)f(or)i(c)m(hec)m(k)i(out)f(the) 227 2757 y(conference's)g(new)d(w)m(ebsite)i(at)g (computationalcomplexit)m(y)-8 b(.org.)141 2941 y(Ketan)32 b(Mulm)m(uley)f(and)g(Milind)e(Sohoni)h(ha)m(v)m(e)j(tak)m(en)g(an)f (algebraic)f(geometry)i(approac)m(h)g(to)f(separating)0 3054 y(complexit)m(y)e(classes.)41 b(In)30 b(this)f(column)g(Ken)h (Regan)h(deciphers)e(this)g(approac)m(h)i(for)f(computer)g(scien)m (tists.)227 3467 y Fn(Understanding)45 b(the)f(Mulm)l(uley-Sohoni)j (Approac)l(h)e(to)d(P)i(vs.)e(NP)1547 3676 y Fr(Kenneth)30 b(W.)h(Regan)2313 3643 y Fm(1)1534 3789 y Fr(Univ)m(ersit)m(y)e(at)i (Bu\013alo)0 4029 y Fl(Abstract)91 b Fr(W)-8 b(e)24 b(explain)d(the)i (essence)h(of)f(K.)g(Mulm)m(uley)e(and)i(M.)g(Sohoni,)g(\\Geometric)h (Complexit)m(y)e(Theory)0 4142 y(I:)k(An)f(Approac)m(h)g(to)i(the)e(P)h (vs.)f(NP)h(and)f(Related)h(Problems")e([MS02)q(])i(for)f(a)h(general)g (complexit)m(y-theory)g(au-)0 4254 y(dience.)39 b(W)-8 b(e)27 b(ev)-5 b(aluate)27 b(the)g(p)s(o)m(w)m(er)f(and)f(prosp)s(ects) h(of)h(the)f(new)g(approac)m(h.)39 b(The)26 b(emphasis)f(is)g(not)i(on) f(probing)0 4367 y(the)36 b(deep)g(mathematics)h(that)g(underlies)c (this)j(w)m(ork,)i(but)d(rather)h(on)g(helping)e(computational)i (complexit)m(y)0 4480 y(theorists)30 b(not)h(v)m(ersed)f(in)f(its)h (bac)m(kground)g(to)h(understand)e(the)h(com)m(binatorics)g(in)m(v)m (olv)m(ed.)0 4766 y Fk(1)135 b(In)l(tro)t(duction)0 4969 y Fr(Consider)28 b(a)i(group)g Fj(G)f Fr(of)h Fj(n)18 b Fo(\002)h Fj(n)30 b Fr(matrices)g Fj(A)f Fr(and)h(a)g(v)m(ector)h Fj(v)i Fr(in)c(an)h Fj(n)p Fr(-dimensional)d(v)m(ector)k(space)f Fj(V)50 b Fr(o)m(v)m(er)32 b(a)0 5082 y(\014eld)g Fj(F)13 b Fr(.)51 b(The)33 b Fi(orbit)42 b Fj(Gv)37 b Fr(is)32 b(the)i(set)g(of)g(images)g Fj(Av)j Fr(o)m(v)m(er)e(all)d Fj(A)f Fo(2)f Fj(G)p Fr(.)50 b(W)-8 b(e)35 b(ask,)f(ho)m(w)g(\\nice")g (a)g(subset)f(of)h Fj(V)0 5195 y Fr(do)s(es)f Fj(Gv)k Fr(form?)50 b(Do)s(es)35 b(it)e(swing)f(arbitrarily)f(close)k(to)f (zero?)52 b(If)33 b(not,)i(and/or)f(if)e(it)i(remains)e(nice)i(when)e (w)m(e)p 0 5254 1560 4 v 104 5308 a Fh(1)138 5340 y Fg(Supp)r(orted)25 b(in)g(part)h(b)n(y)f(NSF)g(gran)n(t)g(CCR-9821040)p eop %%Page: 2 2 2 1 bop 0 91 a Fr(replace)34 b Fj(V)54 b Fr(b)m(y)34 b(the)h Fi(pr)-5 b(oje)g(ctive)37 b(sp)-5 b(ac)g(e)37 b Fj(V)1434 58 y Ff(\003)1474 91 y Fr(|then)c(one)i(sa)m(ys)f(that)h (the)f(action)h(of)f Fj(G)g Fr(on)g Fj(v)j Fr(is)c Fi(stable)7 b Fr(.)52 b(W)-8 b(e)36 b(can)0 204 y(ask)29 b(similar)d(questions)i (for)h(other)g(kinds)e(of)i Fi(gr)-5 b(oup)32 b(actions)38 b Fj(\013)25 b Fr(:)h Fj(G)17 b Fo(\002)f Fj(V)46 b Fo(!)25 b Fj(V)49 b Fr(b)s(esides)27 b Fj(A;)15 b(v)29 b Fo(7!)d Fj(Av)s Fr(,)j(including)0 317 y(cases)i(where)f(the)h(dimension)d(of)i Fj(V)51 b Fr(is)29 b(m)m(uc)m(h)h(larger)h(than)f Fj(n)p Fr(.)141 430 y(Stabilit)m(y)23 b(is)h(informally)d(a)k(notion)f(of)h (not)g(b)s(eing)e(\\c)m(haotic,")28 b(and)c(has)h(dev)m(elop)s(ed)f(in) m(to)g(a)h(ma)5 b(jor)25 b(branc)m(h)f(of)0 543 y(algebraic)31 b(geometry)h(under)e(the)h(guiding)e(in\015uence)h(of)h(D.A.)h(Mumford) e(among)i(others.)43 b(Ketan)32 b(Mulm)m(uley)0 656 y(and)37 b(Milind)e(Sohoni)i([MS02)q(])h(observ)m(e)g(that)h(man)m(y)f (questions)f(ab)s(out)h(complexit)m(y)f(classes)h(can)g(b)s(e)g (re-cast)0 769 y(as)h(questions)e(ab)s(out)i(the)f(nature)h(of)f(group) g(actions)h(on)g(certain)f(v)m(ectors)i(in)d(certain)i(spaces)g(that)g (enco)s(de)0 882 y(problems)g(in)g(these)i(classes.)72 b(This)39 b(surv)m(ey)h(explains)f(their)h(framew)m(ork)g(from)h(a)g (la)m(y)f(p)s(oin)m(t)g(of)h(view,)i(and)0 995 y(attempts)e(to)g(ev)-5 b(aluate)40 b(whether)g(this)f(approac)m(h)h(truly)f(adds)g(new)g(p)s (o)m(w)m(er)h(to)h(attac)m(ks)h(on)e(the)g(P)-8 b(.)41 b(vs.)f(NP)0 1108 y(question.)0 1394 y Fk(2)135 b(Key)45 b(geometric)i(and)d(algebraic)j(concepts)0 1597 y Fr(First)33 b(w)m(e)i(need)f(to)h(de\014ne)e(\\nice.")53 b(Giv)m(en)34 b(a)g(\014eld)f Fj(F)13 b Fr(,)35 b(tak)m(e)h Fj(V)54 b Fr(to)35 b(b)s(e)f(the)g(\014nite-dimensional)d(v)m(ector)k(space)0 1710 y Fj(F)71 1674 y Fe(n)118 1710 y Fr(.)41 b(Also)30 b(write)f Fj(F)13 b Fr([)p Fj(x)769 1724 y Fm(1)809 1710 y Fj(;)i(:)g(:)g(:)h(;)f(x)1062 1724 y Fe(n)1110 1710 y Fr(])30 b(for)g(the)g Fi(ring)39 b Fr(of)30 b(p)s(olynomials)e(in)g Fj(n)i Fr(v)-5 b(ariables)29 b(with)g(co)s(e\016cien)m(ts)i(in)e Fj(F)13 b Fr(.)40 b(The)0 1823 y(solution)29 b(space)j Fj(S)j Fr(of)c(a)h(\014nite)d(set)j(of)f(p)s(olynomial)d(equations)i Fj(p)2262 1837 y Fm(1)2302 1823 y Fr(\()p Fj(x)2389 1837 y Fm(1)2428 1823 y Fj(;)15 b(:)g(:)g(:)i(;)e(x)2682 1837 y Fe(n)2729 1823 y Fr(\))27 b(=)e(0)p Fj(;)15 b(:)g(:)g(:)i(p)3140 1837 y Fe(s)3177 1823 y Fr(\()p Fj(x)3264 1837 y Fm(1)3304 1823 y Fj(;)e(:)g(:)g(:)h(;)f(x)3557 1837 y Fe(n)3605 1823 y Fr(\))26 b(=)g(0)31 b(is)0 1936 y(then)j(a)g(subset)g(of)g Fj(V)20 b Fr(,)35 b(and)f(w)m(e)g(call)g Fj(S)39 b Fr(a)34 b Fi(b)-5 b(asic)37 b(close)-5 b(d)37 b(set)9 b Fr(.)52 b(Finite)33 b(unions)f(and)h(arbitrary)g(in)m(tersections)h(of)0 2049 y(basic)28 b(closed)h(sets)g(form)f(the)h Fi(close)-5 b(d)32 b(sets)k Fr(of)29 b(the)g Fi(Zariski)i(top)-5 b(olo)g(gy)40 b Fr(on)29 b Fj(V)20 b Fr(.)40 b(Their)27 b(complemen)m(ts)i(are)g(Zariski-)0 2161 y Fi(op)-5 b(en)7 b Fr(.)41 b(Prop)s(er)29 b(Z\(ariski\)-closed)f(subsets)g(of)i Fj(V)49 b Fr(are)30 b(\\nice")f Fi(algebr)-5 b(aic)33 b(sets)7 b Fr(.)41 b(Less)29 b(nice)g(are)g(subsets)g(that)h(are)0 2274 y(op)s(en,)e(or)h(are)g(the)g(in)m(tersection)f(of)h(a)g(closed)f (set)h(and)f(an)g(op)s(en)g(set|these)i(are)f(called)e Fi(lo)-5 b(c)g(al)5 b(ly)33 b(close)-5 b(d)9 b Fr(.)42 b(Finite)0 2387 y(unions)32 b(of)i(lo)s(cally-closed)f(sets)i(are)f (the)g(same)h(as)f(all)f(\014nite)g(Bo)s(olean)i(com)m(binations)e(of)h (closed)g(sets|these)0 2500 y(are)d(called)e Fi(c)-5 b(onstructible)34 b(sets)7 b Fr(.)141 2613 y(Ev)m(ery)35 b(Z-closed)f(set)h(is)e(closed)h(in)f(the)h(familiar)e(\\Euclidean")i (top)s(ology)-8 b(,)36 b(but)e(not)g(con)m(v)m(ersely)-8 b(,)37 b(b)s(ecause)0 2726 y(ev)m(ery)28 b(Z-closed)f(set)h(other)f (than)g Fj(V)48 b Fr(itself)26 b(is)g(at)i(most)g(\()p Fj(n)14 b Fo(\000)g Fr(1\)-dimensional.)38 b(Th)m(us)26 b(ev)m(ery)i(Z-closed)f(set)h(other)0 2839 y(than)38 b Fj(V)58 b Fr(has)39 b(measure)f(zero)h(on)f Fj(V)20 b Fr(,)41 b(and)c(b)s(eing)g(Z-op)s(en)h(giv)m(es)g(a)h(particularly)d (strong)i(notion)g(of)g(\\almost)0 2952 y(ev)m(erywhere.")56 b(Just)34 b(as)i(in)e(the)h(Euclidean)e(top)s(ology)-8 b(,)38 b(ev)m(ery)e(set)f Fj(R)f Fo(\022)f Fj(V)56 b Fr(has)35 b(a)g Fi(closur)-5 b(e)3272 2929 y Fr(\026)3252 2952 y Fj(R)36 b Fr(in)e(the)h(Zariski)0 3065 y(top)s(ology)-8 b(,)36 b(de\014ned)c(to)i(b)s(e)f(the)h(in)m(tersection)g(of)g(all)e (Z-closed)i(sets)g(that)g(con)m(tain)g Fj(R)q Fr(.)50 b(It)34 b(is)f(also)h(de\014nable)e(as)20 3155 y(\026)0 3178 y Fj(R)h Fr(=)g Fo(V)7 b Fr(\()p Fo(I)g Fr(\()p Fj(R)q Fr(\)\),)37 b(where)d Fo(I)7 b Fr(\()p Fj(R)q Fr(\))34 b(is)g(the)h(set)h(of)f(all)f Fj(n)p Fr(-ary)g(p)s(olynomials) e(that)k(v)-5 b(anish)33 b(on)i Fj(R)q Fr(,)h(and)e Fo(V)7 b Fr(\()p Fo(I)g Fr(\))35 b(means)0 3291 y(the)f(set)g(of)g(common)g (zero)s(es)g(of)g(those)h(p)s(olynomials.)48 b(Here)34 b Fo(I)j Fr(=)30 b Fo(I)7 b Fr(\()p Fj(R)q Fr(\))34 b(forms)f(an)g Fi(ide)-5 b(al)9 b Fr(,)37 b(meaning)c(that)h(for)0 3403 y(all)f Fj(p;)15 b(q)35 b Fo(2)d(I)41 b Fr(and)34 b(arbitrary)f(p)s (olynomial)e Fj(\013)p Fr(,)37 b Fj(\013p)23 b Fr(+)f Fj(q)35 b Fo(2)d(I)7 b Fr(.)52 b(Not)35 b(all)f(ideals)f(ha)m(v)m(e)j (the)e(form)g Fo(I)7 b Fr(\()p Fj(R)q Fr(\))34 b(for)h(some)0 3516 y Fj(R)q Fr(|those)c(that)g(do)f(are)h Fi(r)-5 b(adic)g(al)9 b Fr(,)34 b(meaning)29 b(that)i(whenev)m(er)g(some)g(p)s(o)m(w)m(er)f Fj(p)2700 3483 y Fe(e)2767 3516 y Fr(of)h(a)g(p)s(olynomial)d Fj(p)i Fr(b)s(elongs)f(to)0 3629 y Fo(I)7 b Fr(,)27 b Fj(p)f Fr(itself)f(b)s(elongs)g(to)j Fo(I)7 b Fr(.)38 b Fi(Hilb)-5 b(ert's)29 b(Nul)5 b(lstel)g(lensatz)39 b Fr(sa)m(ys)27 b(that)g(if)e(the)i(\014eld)e Fj(F)39 b Fr(is)26 b(algebraically)f(closed,)i(then)0 3742 y(there)39 b(is)e(a)i(1-1)h(corresp)s(ondence)e(b)s(et)m(w)m(een)h(Z-closed)f (subsets)g(of)h Fj(V)58 b Fr(and)38 b(radical)f(ideals)h(in)f Fj(F)13 b Fr([)p Fj(x)3509 3756 y Fm(1)3549 3742 y Fj(;)i(:)g(:)g(:)h (;)f(x)3802 3756 y Fe(n)3849 3742 y Fr(].)0 3855 y(Ev)m(ery)28 b(ideal)e(of)h(p)s(olynomials)e Fj(I)34 b Fr(is)26 b Fi(\014nitely)31 b(gener)-5 b(ate)g(d)9 b Fr(,)30 b(meaning)c(that)i (there)g(exist)f Fj(p)3013 3869 y Fm(1)3052 3855 y Fj(;)15 b(:)g(:)g(:)i(;)e(p)3300 3869 y Fe(s)3362 3855 y Fo(2)25 b(I)33 b Fr(suc)m(h)27 b(that)0 3968 y Fo(I)k Fr(comprises)24 b(all)g(the)h(\\algebraic)g(consequences")h Fj(\013)1834 3982 y Fm(1)1874 3968 y Fj(p)1920 3982 y Fm(1)1968 3968 y Fr(+)9 b Fj(:)15 b(:)g(:)c Fr(+)e Fj(\013)2302 3982 y Fe(s)2339 3968 y Fj(p)2385 3982 y Fe(s)2446 3968 y Fr(o)m(v)m(er)26 b(all)e(p)s(olynomials)e Fj(\013)3314 3982 y Fm(1)3354 3968 y Fj(;)15 b(:)g(:)g(:)h(;)f(\013)3613 3982 y Fe(s)3676 3968 y Fr(of)25 b(the)0 4081 y(equations)33 b Fj(p)456 4095 y Fe(i)484 4081 y Fr(.)50 b(The)33 b(Nullstellensatz)f (implies)f(that)j(the)g(equations)f(are)h(unsolv)-5 b(able)32 b(o)m(v)m(er)j(the)e(algebraically)0 4194 y(closed)41 b(\014eld)g(i\013)f(1)i(is)f(a)h(consequence.)75 b(The)41 b(Z-closure)g(of)h(an)f(arbitrary)g(p)s(oin)m(t)f(set)i Fj(R)h Fr(ma)m(y)f(ballo)s(on)e(out)0 4307 y(a)c(lot)g(further)f(than)h (its)f(Euclidean)f(closure|for)h(instance,)i(an)m(y)g(\\op)s(en)e (ball")g(\(of)i(full)c(dimension\))h(in)h(the)0 4420 y(Euclidean)28 b(top)s(ology)j(closes)g(out)f(to)h(all)f(of)g Fj(V)51 b Fr(in)29 b(the)h(Zariski)f(top)s(ology)-8 b(.)141 4533 y(The)36 b(Zariski)e(top)s(ology)j(ma)m(y)g(also)g(b)s(e)e (de\014ned)h(on)g Fi(pr)-5 b(oje)g(ctive)39 b(sp)-5 b(ac)g(e)7 b Fr(,)40 b(and)c(there)g(it)g(has)g(an)h(imp)s(ortan)m(t)0 4645 y(sp)s(ecial-case)j(connection)g(with)e(the)i(familiar)e(top)s (ology)-8 b(.)70 b(De\014ne)40 b Fj(P)13 b Fr(\()p Fj(V)20 b Fr(\))40 b(to)h(b)s(e)e(the)h(space)h(of)f(equiv)-5 b(alence)0 4758 y(classes)42 b(of)h Fj(V)63 b Fr(under)41 b(scalar)h(m)m(ultiplication|i.e.,)h(of)f(one-dimensional)f(linear)f (subspaces.)77 b(This)40 b(is)i(the)0 4871 y Fi(pr)-5 b(oje)g(ctive)38 b(sp)-5 b(ac)g(e)43 b Fr(asso)s(ciated)36 b(to)f Fj(V)21 b Fr(.)54 b(If)35 b(eac)m(h)h(p)s(olynomial)c Fj(p)2189 4885 y Fe(i)2252 4871 y Fr(in)i Fo(f)15 b Fj(p)2469 4885 y Fm(1)2508 4871 y Fj(;)g(:)g(:)g(:)i(;)e(p)2756 4885 y Fe(s)2808 4871 y Fo(g)35 b Fr(is)f Fi(homo)-5 b(gene)g(ous)7 b Fr(,)40 b(meaning)0 4984 y(that)j(all)d(its)i (monomials)e(in)h(unfactored)h(form)f(ha)m(v)m(e)j(the)e(same)g(degree) h Fj(d)2730 4998 y Fe(i)2800 4984 y Fr(\(whic)m(h)e(can)h(b)s(e)g (di\013eren)m(t)f(for)0 5097 y(di\013eren)m(t)f Fj(i)p Fr(\),)j(then)d(their)g(solution)f(space)h Fj(S)46 b Fr(is)39 b(in)m(v)-5 b(arian)m(t)40 b(under)e(scalar)j(m)m(ultiples.)68 b(Th)m(us)39 b Fj(S)45 b Fr(b)s(ecomes)c(a)0 5210 y(subset)29 b(of)i(pro)5 b(jectiv)m(e)30 b(space)h(\(ignoring)d(the)j(zero)f(v)m (ector\),)j(and)c(is)g(a)i(closed)e(set)i(in)e(the)h(Zariski)e(top)s (ology)i(on)0 5323 y Fj(P)13 b Fr(\()p Fj(V)21 b Fr(\).)46 b(The)31 b(Z-closure)h(of)g(an)g(arbitrary)f Fj(R)e Fo(\022)f Fj(P)13 b Fr(\()p Fj(V)20 b Fr(\))33 b(is)e(de\014ned)g(analogously)-8 b(.)46 b(The)31 b(k)m(ey)i(fact)g(is)e(that)i(when)p eop %%Page: 3 3 3 2 bop 0 91 a Fj(V)52 b Fr(is)32 b(a)g(v)m(ector)i(space)f(o)m(v)m(er) g(an)g(algebraically)d(closed)i(\014eld,)g(and)g Fj(S)37 b Fr(is)31 b(a)i Fi(c)-5 b(onstructible)40 b Fr(subset)31 b(of)i Fj(P)13 b Fr(\()p Fj(V)20 b Fr(\),)34 b Fi(the)0 204 y(Z-closur)-5 b(e)33 b(of)g Fj(S)k Fi(c)-5 b(oincides)33 b(with)h(its)f(classic)-5 b(al)34 b(closur)-5 b(e)38 b Fr(\(see)31 b([Mum76)q(]\).)0 448 y Fd(2.1)112 b(Group)38 b(actions)0 619 y Fr(A)45 b Fi(gr)-5 b(oup)47 b(action)52 b Fr(of)45 b(a)g(group)f Fj(G)g Fi(on)52 b Fr(a)45 b(set)g Fj(S)50 b Fr(is)43 b(a)i(mapping)e Fj(\013)50 b Fr(:)f Fj(G)29 b Fo(\002)h Fj(S)54 b Fo(!)49 b Fj(S)g Fr(that)c(con)m(v)m (erts)h(group)0 732 y(m)m(ultiplication)33 b(in)m(to)i(comp)s(osition:) 51 b(for)35 b(all)g Fj(g)s(;)15 b(h)35 b Fo(2)f Fj(G)h Fr(and)h Fj(u)e Fo(2)g Fj(S)5 b Fr(,)37 b Fj(\013)p Fr(\()p Fj(g)s(h;)15 b(u)p Fr(\))37 b(=)d Fj(\013)p Fr(\()p Fj(g)s(;)15 b(\013)p Fr(\()p Fj(h;)g(u)p Fr(\)\).)61 b(If)35 b Fj(e)h Fr(is)f(the)0 845 y(iden)m(tit)m(y)f(of)h Fj(G)p Fr(,)g(it)f(is)f (customary|but)i(not)f(alw)m(a)m(ys)h(necessary|to)h(stipulate)d(that)j Fj(\013)p Fr(\()p Fj(e;)15 b(u)p Fr(\))34 b(=)e Fj(u)i Fr(for)g(all)g Fj(u)p Fr(.)0 958 y(The)40 b(canonical)g(example)g(is)f (when)g Fj(S)47 b Fr(=)42 b Fj(V)62 b Fr(=)41 b Fl(C)1839 925 y Fe(n)1926 958 y Fr(and)f Fj(G)f Fr(is)h(a)g(group)g(of)h(complex) f Fj(n)26 b Fo(\002)g Fj(n)40 b Fr(matrices)h Fj(A)p Fr(,)0 1071 y(with)33 b Fj(\013)p Fr(\()p Fj(A;)15 b(u)p Fr(\))33 b(=)e Fj(Au)p Fr(.)51 b(Then)33 b Fj(\013)p Fr(\()p Fj(AB)5 b(;)15 b(u)p Fr(\))33 b(=)e Fj(AB)5 b(u)33 b Fr(since)h(m)m(ultiplication)d(is)i(asso)s(ciativ)m(e.)52 b(Another)34 b(action)g(is)0 1184 y Fj(\013)58 1151 y Ff(0)82 1184 y Fr(\()p Fj(A;)15 b(u)p Fr(\))31 b(=)f Fj(AuA)632 1151 y Ff(\000)p Fm(1)727 1184 y Fr(;)35 b(note)f(that)g Fj(\013)1250 1151 y Ff(0)1274 1184 y Fr(\()p Fj(AB)5 b(;)15 b(u)p Fr(\))31 b(=)f(\()p Fj(AB)5 b Fr(\))p Fj(u)p Fr(\()p Fj(AB)g Fr(\))2186 1151 y Ff(\000)p Fm(1)2312 1184 y Fr(=)30 b Fj(AB)5 b(uB)2681 1151 y Ff(\000)p Fm(1)2774 1184 y Fj(A)31 b Fr(=)f Fj(\013)3032 1151 y Ff(0)3056 1184 y Fr(\()p Fj(A;)15 b(\013)3257 1151 y Ff(0)3281 1184 y Fr(\()p Fj(B)5 b(;)15 b(u)p Fr(\)\).)51 b(This)32 b(is)0 1297 y(subsumed)c(b)m(y)i(the)h(action)g(of)f(the)h(pro)s(duct)e (group)h Fj(G)1907 1264 y Ff(0)1955 1297 y Fr(=)25 b Fj(G)20 b Fo(\002)f Fj(G)30 b Fr(giv)m(en)h(b)m(y)f Fj(\013)2757 1311 y Fm(2)2796 1297 y Fr(\(\()p Fj(A;)15 b(B)5 b Fr(\))p Fj(;)15 b(u)p Fr(\))28 b(=)d Fj(AuB)3528 1264 y Ff(\000)p Fm(1)3622 1297 y Fr(.)141 1410 y(Where)k(the)h(action)f(referred)f(to)i (is)e(clear,)h(it)g(is)f(written)g(simply)e(with)i(pro)s(duct)g (notation,)i(namely)e Fj(g)6 b Fo(\001)r Fj(u)29 b Fr(in)0 1523 y(place)h(of)h Fj(\013)p Fr(\()p Fj(g)s(;)15 b(u)p Fr(\).)43 b(The)30 b(action)g(is)g(said)f(to)i(\\giv)m(e)h Fj(S)j Fr(the)30 b(structure)g(of)h(a)g Fj(G)p Fr(-mo)s(dule.")141 1635 y(The)k(action)i(used)e(most)h(critically)e(b)m(y)h(Mulm)m(uley)f (and)i(Sohoni)e(tak)m(es)j Fj(G)e Fr(to)i(b)s(e)e(a)h(group)g(of)g(in)m (v)m(ertible)0 1748 y Fj(m)27 b Fo(\002)f Fj(m)40 b Fr(matrices)g Fj(B)45 b Fr(and)39 b Fj(h)i Fr(to)g(b)s(e)f(an)g Fj(m)p Fr(-v)-5 b(ariable)39 b(p)s(olynomial)e(that)k(is)e(homogeneous)i(of)g (some)f(degree)0 1861 y Fj(d)j Fo(\024)f Fj(m)p Fr(.)71 b(Suc)m(h)40 b Fj(h)h Fr(b)s(elong)f(to)h(the)g(v)m(ector)h(space)f Fo(V)1883 1876 y Fe(d)1964 1861 y Fr(o)m(v)m(er)h Fl(C)e Fr(of)h(dimension)d Fj(D)45 b Fr(=)d(\()3122 1823 y Fe(m)p Fm(+)p Fe(d)p Ff(\000)p Fm(1)3123 1889 y Fe(d)3386 1861 y Fr(\),)i(with)39 b(basis)0 1974 y(giv)m(en)e(b)m(y)f(the)h(set)g(of)g (monomials)e(of)i(degree)g Fj(d)p Fr(.)59 b(The)36 b(action)h(is)f Fj(\013)2416 1988 y Fe(G)2475 1974 y Fr(\()p Fj(B)5 b(;)15 b(h)p Fr(\))37 b(=)e Fj(h)2906 1941 y Ff(0)2930 1974 y Fr(,)j(where)e Fj(h)3314 1941 y Ff(0)3374 1974 y Fr(is)g(de\014ned)f (for)0 2087 y(all)e Fj(v)j Fo(2)31 b Fl(C)378 2054 y Fe(m)479 2087 y Fr(b)m(y)j Fj(h)661 2054 y Ff(0)685 2087 y Fr(\()p Fj(v)s Fr(\))f(=)f Fj(h)p Fr(\()p Fj(v)s(B)1146 2054 y Ff(\000)p Fm(1)1241 2087 y Fr(\).)53 b(Again,)36 b(if)d Fj(C)41 b Fr(then)34 b(sends)g Fj(h)2354 2054 y Ff(0)2412 2087 y Fr(to)h Fj(h)2579 2054 y Ff(00)2654 2087 y Fr(=)c Fj(\025w)r(:h)2953 2054 y Ff(0)2978 2087 y Fr(\()p Fj(w)r(C)3152 2054 y Ff(\000)p Fm(1)3247 2087 y Fr(\),)36 b(then)e(for)g(all)f Fj(v)s Fr(,)0 2200 y Fj(h)52 2167 y Ff(00)95 2200 y Fr(\()p Fj(v)s Fr(\))26 b(=)f Fj(h)p Fr(\()p Fj(v)s(C)540 2167 y Ff(\000)p Fm(1)635 2200 y Fj(B)709 2167 y Ff(\000)p Fm(1)803 2200 y Fr(\),)31 b(whic)m(h)e(is)g(where)h Fj(B)5 b(C)36 b Fr(sends)30 b Fj(h)p Fr(.)0 2413 y Fl(De\014nition)35 b(2.1)h(\(see)e(discussions)j (in)e([MS02)q(]\).)45 b Fr(A)25 b(p)s(olynomial)d Fj(h)j Fr(is)f Fi(stable)32 b Fr(under)24 b(the)h Fj(G)f Fr(action)h(if)f(the) 0 2525 y(orbit)34 b Fj(Gh)g Fr(in)f Fo(V)42 b Fr(is)34 b(Z-closed,)h(and)f Fi(semi-stable)42 b Fr(if)34 b(the)h(Z-closure)f (of)g(the)h(orbit)f Fj(Gh)g Fr(in)g Fo(V)41 b Fr(do)s(es)35 b(not)g(con)m(tain)0 2638 y(the)d(zero)h(p)s(olynomial.)42 b(Otherwise)30 b(it)i(is)e Fi(unstable)40 b Fr(or)31 b Fi(nilp)-5 b(otent)9 b Fr(.)47 b(The)31 b(set)h(of)g(unstable)f(p)s (olynomials)e(is)i(the)0 2751 y Fi(nul)5 b(l)33 b(c)-5 b(one)37 b Fr(of)31 b Fo(V)7 b Fr(.)141 2864 y(These)31 b(attributes)f(are)i(unc)m(hanged)e(on)h(m)m(ultiplying)c Fj(h)32 b Fr(b)m(y)e(an)m(y)i(scalar,)f(so)g(they)g(can)g(also)g(b)s(e) f(applied)f(to)0 2977 y(p)s(oin)m(ts)g(in)g(the)i(pro)5 b(jectiv)m(e)31 b(space)g Fj(P)13 b Fr(\()p Fo(V)7 b Fr(\).)0 3190 y(Orbits)36 b(in)g(a\016ne)i(space)g(are)g(alw)m(a)m(ys)g (lo)s(cally)e(closed)h(\(see)h(section)g(8.3)h(in)d(the)i(text)g ([Hum81)q(]\),)i(and)d(hence)0 3303 y(constructible,)c(but)g(need)f (not)i(b)s(e)f(closed.)49 b(The)32 b Fi(b)-5 b(oundary)43 b Fr(of)34 b(the)f(orbit,)h(namely)e(its)h(Z-closure)f(min)m(us)g(the)0 3415 y(orbit,)i(is)e(also)i(preserv)m(ed)f(b)m(y)g Fj(G)p Fr(,)h(so)g(it)f(is)f(a)i(union)e(of)h(orbits.)49 b(Since)33 b(in)f(pro)5 b(jectiv)m(e)34 b(space)g(this)e(is)h(the)h(same)0 3528 y(as)d(the)g(classical)f(b)s(oundary)-8 b(,)30 b(it)g(is)g(lo)m(w) m(er-dimensional)f(than)i(the)g(original)e(orbit,)h(so)h(all)f(orbits)g (in)g(the)h(union)0 3641 y(are)d(lo)m(w)m(er-dimensional.)37 b(Iterating)27 b(this)f(reason)i(yields)d(the)j(existence)f(of)h (orbits)e(of)h(minim)m(um)e(dimension)f(in)0 3754 y(these)31 b(b)s(oundaries)d(that)j(themselv)m(es)f(are)h(closed,)f(so)h(stable)f (p)s(oin)m(ts)f(alw)m(a)m(ys)i(exist.)141 3867 y(Not)37 b(to)g(b)s(e)e(confused)g(with)g(\\stable")h(are)h(the)f(follo)m(wing)e (k)m(ey)j(concepts.)58 b(The)35 b Fi(isotr)-5 b(opy)41 b(sub)-5 b(gr)g(oup)42 b Fr(of)37 b(a)0 3980 y(v)m(ector)j Fj(v)i Fo(2)c Fj(V)59 b Fr(under)37 b(a)i(giv)m(en)f(action)h(b)m(y)f (a)h(group)f Fj(G)g Fr(is)f(de\014ned)h(b)m(y)g Fj(G)2675 3994 y Fe(v)2754 3980 y Fr(=)g Fo(f)15 b Fj(g)43 b Fo(2)c Fj(G)f Fr(:)h Fj(g)s(v)k Fr(=)38 b Fj(v)18 b Fo(g)p Fr(.)66 b(This)0 4093 y(is)41 b(the)h(subgroup)e(of)i(elemen)m(ts)g(whose)g (action)g(lea)m(v)m(es)h Fj(v)i Fr(\014xed.)74 b(The)41 b(orbit)g(is)g(then)g(the)h(singleton)f Fo(f)15 b Fj(v)k Fo(g)p Fr(,)0 4206 y(whic)m(h)40 b(is)h(closed,)j(so)e(suc)m(h)g Fj(v)j Fr(are)d(stable.)74 b(The)41 b Fi(stabilizer)52 b Fr(of)42 b(a)g(v)m(ector)h(subspace)e Fj(W)54 b Fr(of)42 b Fj(V)62 b Fr(is)40 b(giv)m(en)i(b)m(y)0 4319 y Fj(G)72 4333 y Fe(W)180 4319 y Fr(=)28 b Fo(f)15 b Fj(g)33 b Fo(2)28 b Fj(G)g Fr(:)h(\()p Fo(8)p Fj(w)h Fo(2)e Fj(W)13 b Fr(\))28 b Fj(g)s(w)k Fo(2)c Fj(W)g Fo(g)p Fr(.)46 b(The)32 b(isotrop)m(y)g(subgroup)f Fj(G)2561 4333 y Fe(w)2649 4319 y Fr(is)g(also)h(called)g(the)g(\\stabilizer")g(of)0 4432 y Fj(w)r Fr(,)g(but)e(w)m(e)i(try)f(to)g(minimize)e(the)i(usage)h (of)f(\\stable.)16 b(.)f(.)h(")42 b(and)31 b(prefer)f(to)i(sa)m(y)g Fj(w)h Fr(is)d Fi(\014xe)-5 b(d)41 b Fr(b)m(y)31 b Fj(G)3389 4446 y Fe(w)3445 4432 y Fr(.)43 b(Note)32 b(that)0 4545 y(stabilizing)c Fj(W)43 b Fr(is)29 b(w)m(eak)m(er)j(than)e(\014xing)f (ev)m(ery)i Fj(w)d Fo(2)d Fj(W)13 b Fr(,)30 b(when)f(w)m(e)i(sa)m(y)g Fj(W)43 b Fr(is)29 b Fi(p)-5 b(ointwise)35 b(\014xe)-5 b(d)9 b Fr(.)0 4757 y Fl(Example)34 b(2.1.)46 b Fr(This)29 b(expands)g(\\5.3.1)k(Example)d(2")i(in)d([MS02)q(],)i(itself)f(ascrib) s(ed)f(to)i([PV91)r(].)41 b(Let)32 b Fj(V)46 b Fr(=)25 b Fl(C)3861 4724 y Fm(4)0 4870 y Fr(b)s(e)h(though)m(t)h(of)g(as)g(the) g(space)g(of)g(homogeneous)g(p)s(olynomials)d(of)j(degree)g(3)g(via)g (co)s(e\016cien)m(ts)g(of)g(the)g(four)f(basic)0 4983 y(monomials)i Fj(x)506 4950 y Fm(3)545 4983 y Fj(;)15 b(x)637 4950 y Fm(2)677 4983 y Fj(y)s(;)g(xy)865 4950 y Fm(2)904 4983 y Fj(;)g(y)992 4950 y Fm(3)1032 4983 y Fr(.)40 b(T)-8 b(ak)m(e)31 b Fj(G)24 b Fr(=)h Fi(SL)1616 4997 y Fm(2)1656 4983 y Fr(\()p Fl(C)p Fr(\),)30 b Fj(f)k Fr(=)25 b Fj(x)2084 4950 y Fm(2)2123 4983 y Fj(y)s Fr(,)30 b(and)e Fj(g)h Fr(=)c Fj(x)2621 4950 y Fm(3)2678 4983 y Fr(+)17 b Fj(y)2814 4950 y Fm(3)2854 4983 y Fr(.)40 b(Then)28 b Fj(f)38 b Fr(is)28 b(iden)m(ti\014ed)f(with)0 5096 y(\(0)p Fj(;)15 b Fr(1)p Fj(;)g Fr(0)p Fj(;)g Fr(0\),)40 b(and)34 b Fj(g)k Fr(with)c(\(1)p Fj(;)15 b Fr(0)p Fj(;)g Fr(0)p Fj(;)g Fr(1\).)57 b(T)-8 b(ak)m(e)36 b Fj(A)d Fr(=)f(\()1839 5063 y Fe(a)g(c)1840 5122 y(b)e(d)1958 5096 y Fr(\))1993 5054 y Ff(\000)p Fm(1)2087 5096 y Fr(,)36 b(so)f(that)h(the)f(action)g Fj(A)8 b Fo(\001)g Fj(f)42 b Fr(=)32 b Fj(\025x:f)10 b Fr(\()p Fj(xA)3543 5063 y Ff(\000)p Fm(1)3638 5096 y Fr(\))35 b(giv)m(es)0 5209 y Fj(A)s Fo(\001)s Fj(f)h Fr(=)25 b Fj(f)10 b Fr(\()p Fj(ax)18 b Fr(+)g Fj(by)s(;)d(cx)k Fr(+)g Fj(dy)s Fr(\))26 b(=)e(\()p Fj(ax)19 b Fr(+)g Fj(by)s Fr(\))1517 5176 y Fm(2)1556 5209 y Fr(\()p Fj(cx)h Fr(+)e Fj(dy)s Fr(\))30 b(and)f Fj(A)s Fo(\001)s Fj(g)h Fr(=)25 b(\()p Fj(ax)19 b Fr(+)f Fj(by)s Fr(\))2760 5176 y Fm(3)2818 5209 y Fr(+)h(\()p Fj(cx)g Fr(+)f Fj(dy)s Fr(\))3272 5176 y Fm(3)3312 5209 y Fr(.)41 b(Then)28 b(sub)5 b(ject)p eop %%Page: 4 4 4 3 bop 0 91 a Fr(to)31 b Fj(ad)21 b Fo(\000)f Fj(bc)25 b Fr(=)g(1,)31 b(the)g(orbits)e(of)i Fj(f)39 b Fr(and)30 b Fj(g)k Fr(are:)617 296 y Fj(Gf)92 b Fr(=)83 b Fo(f)15 b Fr(\()p Fj(a)1123 258 y Fm(2)1163 296 y Fj(c;)g Fr(2)p Fj(abc)22 b Fr(+)e Fj(a)1574 258 y Fm(2)1613 296 y Fj(d;)15 b Fr(2)p Fj(abd)22 b Fr(+)e Fj(b)2031 258 y Fm(2)2070 296 y Fj(c;)15 b(bd)2235 258 y Fm(2)2276 296 y Fr(\))26 b(:)f Fj(ad)c Fo(\000)e Fj(bc)26 b Fr(=)f(1)15 b Fo(g)625 433 y Fj(Gg)86 b Fr(=)d Fo(f)15 b Fr(\()p Fj(a)1123 396 y Fm(3)1183 433 y Fr(+)20 b Fj(c)1313 396 y Fm(3)1353 433 y Fj(;)15 b Fr(3)p Fj(a)1486 396 y Fm(2)1526 433 y Fj(b)21 b Fr(+)e(3)p Fj(c)1760 396 y Fm(2)1801 433 y Fj(d;)c Fr(3)p Fj(ab)2020 396 y Fm(2)2081 433 y Fr(+)k(3)p Fj(cd)2302 396 y Fm(2)2343 433 y Fj(;)c(b)2422 396 y Fm(3)2482 433 y Fr(+)20 b Fj(d)2620 396 y Fm(3)2660 433 y Fr(\))25 b(:)h Fj(ad)20 b Fo(\000)g Fj(bc)26 b Fr(=)f(1)15 b Fo(g)0 638 y Fr(T)-8 b(o)38 b(determine)e(the)i(isotrop)m(y)f (subgroup)e Fj(G)1565 653 y Fe(f)1647 638 y Fr(for)i(the)h(action)g(on) f Fj(V)20 b Fr(,)39 b(w)m(e)f(solv)m(e)f Fj(a)2925 605 y Fm(2)2965 638 y Fj(c)g Fr(=)f(0,)k(2)p Fj(abc)26 b Fr(+)e Fj(a)3598 605 y Fm(2)3638 638 y Fj(d)37 b Fr(=)f(1,)0 751 y(2)p Fj(abd)26 b Fr(+)e Fj(b)339 718 y Fm(2)378 751 y Fj(c)37 b Fr(=)f(0,)k(and)c Fj(b)893 718 y Fm(2)933 751 y Fj(d)h Fr(=)f(0,)j(together)g(with)d Fj(ad)25 b Fo(\000)f Fj(bc)37 b Fr(=)f(1.)62 b(W)-8 b(e)38 b(cannot)g(ha)m(v)m(e)g Fj(a)f Fr(=)f(0)i(b)m(y)f(the)g(second)0 863 y(equation,)26 b(so)g Fj(c)g Fr(=)f(0,)i(and)d(then)h(b)m(y)h Fj(ad)10 b Fo(\000)g Fj(bc)26 b Fr(=)f(1)g(w)m(e)h(cannot)g(ha)m(v)m(e)h Fj(d)e Fr(=)g(0)h(either.)39 b(So)25 b Fj(b)g Fr(=)g(0)h(and)f(w)m(e)g (are)h(left)f(to)0 976 y(solv)m(e)31 b Fj(a)271 943 y Fm(2)310 976 y Fj(d)26 b Fr(=)g(1)31 b(and)f Fj(ad)c Fr(=)f(1.)42 b(This)28 b(forces)j Fj(a)26 b Fr(=)f Fj(d)h Fr(=)f(1,)32 b(so)f(w)m(e)g(get)g(the)g(iden)m(tit)m(y)f(matrix)g(only) -8 b(,)30 b(meaning)g(that)0 1089 y Fj(G)72 1104 y Fe(f)142 1089 y Fr(on)25 b Fj(V)45 b Fr(is)24 b(trivial.)37 b Fi(However)10 b Fr(,)26 b(if)e(w)m(e)h(w)m(ork)g(in)f Fj(P)13 b Fr(\()p Fj(V)21 b Fr(\),)26 b(then)f(the)g(second)g(equation) g(b)s(ecomes)g(2)p Fj(abc)9 b Fr(+)g Fj(a)3611 1056 y Fm(2)3652 1089 y Fj(d)25 b Fr(=)g Fj(m)0 1202 y Fr(for)i(a)h(general)g (nonzero)g(m)m(ultiplier)c Fj(m)p Fr(,)29 b(and)e(then)g(w)m(e're)h (left)g(to)g(solv)m(e)g Fj(a)2547 1169 y Fm(2)2587 1202 y Fj(d)d Fr(=)g Fj(m)j Fr(and)f Fj(ad)e Fr(=)g(1.)40 b(This)26 b(is)h(solv)m(ed)0 1315 y(b)m(y)36 b Fj(a)g Fr(=)e Fj(m)p Fr(,)k Fj(d)e Fr(=)f(1)p Fj(=m)p Fr(,)k(giving)c(us)g(a)i (one-dimensional)d(isotrop)m(y)i(subgroup)f(of)i(diagonal)e(matrices.) 59 b(Since)0 1428 y Fj(P)13 b Fr(\()p Fj(V)21 b Fr(\))30 b(is)g(3-dimensional,)e(this)h(implies)f(that)j(the)g(orbit)e Fj(Gf)39 b Fr(is)30 b(t)m(w)m(o-dimensional.)141 1541 y(F)-8 b(or)31 b Fj(G)376 1555 y Fe(g)445 1541 y Fr(in)d Fj(V)20 b Fr(,)30 b(w)m(e)g(solv)m(e)g Fj(a)1081 1508 y Fm(3)1140 1541 y Fr(+)18 b Fj(c)1268 1508 y Fm(3)1333 1541 y Fr(=)25 b(1,)30 b Fj(a)1577 1508 y Fm(2)1617 1541 y Fj(b)19 b Fr(+)f Fj(c)1803 1508 y Fm(2)1843 1541 y Fj(d)25 b Fr(=)g(0,)31 b Fj(ab)2199 1508 y Fm(2)2257 1541 y Fr(+)18 b Fj(cd)2432 1508 y Fm(2)2498 1541 y Fr(=)25 b(0,)30 b(and)f Fj(b)2909 1508 y Fm(3)2967 1541 y Fr(+)19 b Fj(d)3104 1508 y Fm(3)3169 1541 y Fr(=)25 b(1,)30 b(together)h(with)0 1654 y Fj(ad)15 b Fo(\000)g Fj(bc)26 b Fr(=)f(1.)40 b(If)28 b Fj(a)d Fr(=)g(0,)k(then)e(w)m(e)i(get)g Fj(c)1387 1621 y Fm(3)1452 1654 y Fr(=)c(1)j(and)f Fj(c)1834 1621 y Fm(2)1874 1654 y Fj(d)e Fr(=)g(0,)k(so)f Fj(d)e Fr(=)f(0)j(and)f(w)m (e're)i(left)f(with)e Fj(bc)g Fr(=)f Fo(\000)p Fr(1,)j Fj(c)3668 1621 y Fm(3)3733 1654 y Fr(=)d(1,)0 1767 y(and)j Fj(b)214 1734 y Fm(3)279 1767 y Fr(=)c(1.)41 b(This)27 b(is)g(imp)s(ossible)e(b)s(ecause)j(no)h(t)m(w)m(o)g(of)g(the)g(cub)s (e)e(ro)s(ots)i(of)f(unit)m(y)g(m)m(ultiply)d(to)30 b Fo(\000)p Fr(1.)40 b(So)28 b Fj(a)d Fo(6)p Fr(=)g(0,)0 1880 y(and)37 b(symmetrically)-8 b(,)39 b Fj(d)f Fo(6)p Fr(=)f(0.)64 b(Th)m(us)36 b(w)m(e)j(can)f(divide)d(b)m(y)j Fj(a)g Fr(to)g(get)h Fj(b)f Fr(=)f Fo(\000)p Fj(c)2747 1847 y Fm(2)2787 1880 y Fj(d=a)2927 1847 y Fm(2)2967 1880 y Fr(,)j(and)d(then)h Fj(ab)3518 1847 y Fm(2)3595 1880 y Fr(=)f Fo(\000)p Fj(cd)3860 1847 y Fm(2)0 1993 y Fr(simpli\014es)c(to)k Fj(c)557 1960 y Fm(4)597 1993 y Fj(=a)690 1960 y Fm(3)765 1993 y Fr(=)f Fo(\000)p Fj(c)p Fr(.)59 b(If)36 b Fj(c)g Fo(6)p Fr(=)f(0,)j(this)e(giv)m(es)h Fj(c)1904 1960 y Fm(3)1943 1993 y Fj(=a)2036 1960 y Fm(3)2112 1993 y Fr(=)e Fo(\000)p Fr(1,)j(but)e(that)h(con)m(tradicts)g Fj(a)3295 1960 y Fm(3)3359 1993 y Fr(+)24 b Fj(c)3493 1960 y Fm(3)3568 1993 y Fr(=)36 b(1.)59 b(So)0 2105 y Fj(c)33 b Fr(=)g(0,)j(and)f(symmetrically)-8 b(,)35 b Fj(b)e Fr(=)f(0.)55 b(This)33 b(lea)m(v)m(es)j Fj(a)1899 2072 y Fm(3)1972 2105 y Fr(=)c(1,)37 b Fj(d)2229 2072 y Fm(3)2302 2105 y Fr(=)32 b(1,)37 b(and)d Fj(ad)f Fr(=)g(1,)k(so)e Fj(G)3220 2119 y Fe(g)3294 2105 y Fr(is)f(the)h(\014nite)f(set)0 2218 y(\()50 2185 y Fe(!)f Fm(0)51 2241 y(0)j(\026)-41 b Fe(!)182 2218 y Fr(\))30 b(o)m(v)m(er)i(the)f(three)f(cub)s(e)g(ro)s (ots)g Fj(!)k Fr(of)c(unit)m(y)-8 b(.)40 b(Th)m(us)29 b Fj(G)2037 2232 y Fe(g)2107 2218 y Fr(is)h(\014nite.)141 2331 y(No)m(w)c(w)m(e)f(can)g(sho)m(w)f(that)h(no)g(m)m(ultiple)d Fj(mf)34 b Fr(of)25 b Fj(f)34 b Fr(lies)23 b(in)h(the)g(orbit)g Fj(Gg)s Fr(.)39 b(Here)25 b(w)m(e)g(try)g(to)g(solv)m(e)g Fj(a)3501 2298 y Fm(3)3550 2331 y Fr(+)9 b Fj(c)3669 2298 y Fm(3)3733 2331 y Fr(=)25 b(0,)0 2444 y Fj(a)48 2411 y Fm(2)87 2444 y Fj(b)d Fr(+)g Fj(c)280 2411 y Fm(2)319 2444 y Fj(d)30 b Fr(=)e Fj(m)p Fr(,)33 b Fj(ab)720 2411 y Fm(2)782 2444 y Fr(+)21 b Fj(cd)960 2411 y Fm(2)1029 2444 y Fr(=)29 b(0,)k Fj(b)1271 2411 y Fm(3)1333 2444 y Fr(+)21 b Fj(d)1472 2411 y Fm(3)1541 2444 y Fr(=)28 b(0,)34 b(and)e Fj(ad)22 b Fo(\000)g Fj(bc)29 b Fr(=)g(1.)47 b(If)32 b Fj(d)e Fr(=)e(0)33 b(then)g Fj(b)c Fr(=)f(0)33 b(and)f Fj(ad)22 b Fo(\000)g Fj(bc)29 b Fr(=)g(1)0 2557 y(is)39 b(imp)s(ossible;)j(th)m(us)d(w)m(e)i(can)f(divide)e(b)m(y)i Fj(d)p Fr(.)70 b(Substituting)37 b Fj(a)k Fr(=)g(\(1)28 b(+)e Fj(bc)p Fr(\))p Fj(=d)42 b Fr(in)c(the)j(third)d(equation)i(and)0 2670 y(m)m(ultiplying)24 b(through)k(b)m(y)f Fj(d)h Fr(giv)m(es)h Fj(b)1282 2637 y Fm(2)1336 2670 y Fr(+)15 b Fj(b)1461 2637 y Fm(3)1501 2670 y Fj(c)g Fr(+)g Fj(cd)1727 2637 y Fm(3)1793 2670 y Fr(=)25 b(0.)40 b(But)28 b(since)f Fj(b)2436 2637 y Fm(3)2491 2670 y Fr(+)15 b Fj(d)2624 2637 y Fm(3)2689 2670 y Fr(=)25 b(0)j(this)f(giv)m(es)h Fj(b)d Fr(=)g(0)j(and)g Fj(d)d Fr(=)g(0,)0 2783 y(sending)35 b(us)g(bac)m(k)j(to)f(the)f(impasse.)58 b(Th)m(us)35 b(without)h(ev)m(en)h(considering)d(the)j(equation)f(with)f Fj(m)h Fr(w)m(e)h(cannot)0 2896 y(solv)m(e)e(this.)52 b Fi(However)10 b Fr(,)36 b(w)m(e)f(can)g(come)h(within)c(an)m(y)j (desired)e Fj(\017)f(>)g Fr(0)j(of)g Fj(f)43 b Fr(in)34 b(eac)m(h)h(co)s(e\016cien)m(t.)55 b(T)-8 b(ak)m(e)36 b Fj(d)c Fr(=)g(0,)0 3009 y Fj(b)25 b Fr(=)g Fj(\017)p Fr(,)31 b Fj(a)25 b Fr(=)g(1)p Fj(=\017)p Fr(,)31 b(and)f Fj(c)c Fr(=)f Fo(\000)p Fr(1)p Fj(=\017)p Fr(.)41 b(Then)1557 3213 y Fj(Ag)29 b Fr(=)c(\(0)p Fj(;)15 b Fr(3)p Fj(=\017;)g Fr(3)p Fj(\017;)g(\017)2239 3175 y Fm(3)2282 3213 y Fr(\))p Fj(:)0 3417 y Fr(Th)m(us)35 b(the)h(m)m(ultiple)e(\(3)p Fj(=\017)p Fr(\))p Fj(f)47 b Fr(of)36 b Fj(f)46 b Fr(lies)35 b(within)e Fj(\017)j Fr(of)g(the)h(orbit,)g(so)f(in)f(pro)5 b(jectiv)m(e)37 b(space,)h Fj(f)45 b Fr(lies)35 b(within)f(the)0 3530 y(Z-closure)d(of)g(the)g(orbit.)43 b(The)31 b(fact)h(that)g (progressiv)m(ely)e(higher)g(m)m(ultiples)f(of)i Fj(f)41 b Fr(are)31 b(needed)g(in)f(a\016ne)h(space)0 3643 y(is)e(t)m(ypical.) 141 3856 y(A)21 b(simple)d(example)j(of)f(isotrop)m(y)h(is)f(to)h(note) g(that)g(symmetric)f(p)s(olynomials)e(are)j(\014xed)f(under)f(p)s(erm)m (utations)0 3968 y(of)42 b(the)g(v)-5 b(ariables,)44 b(i.e.)75 b(b)m(y)42 b(p)s(erm)m(utation)f(matrices)g(applied)f(to)j (the)f(argumen)m(t)g(v)-5 b(ariables.)74 b(F)-8 b(or)43 b(another)0 4081 y(example,)30 b(note)g(that)g(when)f Fj(A)h Fr(and)f Fj(B)34 b Fr(are)c(in)m(v)m(ertible)e Fj(n)19 b Fo(\002)f Fj(n)29 b Fr(matrices)h(of)g(equal)f(determinan)m (t,)h(and)f Fj(Y)49 b Fr(is)29 b(an)0 4194 y Fj(n)14 b Fo(\002)g Fj(n)26 b Fr(matrix)h(of)g(v)-5 b(ariables,)27 b(the)h(action)f Fj(Y)46 b Fo(7!)25 b Fj(AY)20 b(B)1881 4161 y Ff(\000)p Fm(1)2002 4194 y Fr(preserv)m(es)27 b(the)h(determinan)m(t,)f(since)g(det\()p Fj(AY)21 b(B)3675 4161 y Ff(\000)p Fm(1)3769 4194 y Fr(\))k(=)0 4307 y(det\()p Fj(A)p Fr(\))15 b(det)r(\()p Fj(Y)20 b Fr(\))p Fj(=)15 b Fr(det)q(\()p Fj(B)5 b Fr(\))26 b(=)f(det\()p Fj(Y)20 b Fr(\).)38 b(This)21 b(linear)f(transformation)h(of)h Fj(Y)42 b Fr(can)22 b(b)s(e)g(expressed)f(as)h(a)h(matrix)e Fj(S)3767 4321 y Fe(A;B)0 4420 y Fr(of)i(size)h Fj(n)317 4387 y Fm(2)362 4420 y Fo(\002)6 b Fj(n)494 4387 y Fm(2)555 4420 y Fr(applied)21 b(to)j Fj(Y)44 b Fr(unrolled)20 b(as)k(a)f(v)m(ector.)40 b(This)22 b(matrix)g(has)h(det)q(\()p Fj(S)2754 4434 y Fe(A;B)2887 4420 y Fr(\))j(=)e(1,)i(so)d Fj(S)3299 4434 y Fe(A;B)3457 4420 y Fo(2)i Fi(SL)3651 4440 y Fe(n)3694 4421 y Fh(2)3733 4420 y Fr(\()p Fj(F)13 b Fr(\).)0 4533 y(The)36 b(set)h(of)g(all)e(matrices)i(arising)e(as)h Fj(S)1427 4547 y Fe(A;B)1597 4533 y Fr(with)f Fj(A;)15 b(B)41 b Fr(as)c(ab)s(o)m(v)m(e)h(forms)e(a)h(subgroup)e Fj(R)i Fr(of)f Fi(SL)3474 4553 y Fe(n)3517 4534 y Fh(2)3555 4533 y Fr(\()p Fj(F)14 b Fr(\))36 b(that)0 4646 y(\014xes)30 b(det)331 4660 y Fe(n)378 4646 y Fr(.)141 4759 y(If)j(w)m(e)g(only)f (care)i(ab)s(out)f(preserving)e(det)j(up)e(to)h(scalar)g(m)m (ultiples|i.e.)e(if)h(w)m(e)i(consider)d(det)j(as)f(existing)0 4872 y(in)27 b(pro)5 b(jectiv)m(e)29 b(space|then)g(w)m(e)g(can)f (de\014ne)g Fj(R)1660 4839 y Ff(0)1711 4872 y Fr(without)g(the)h (restriction)e(det\()p Fj(B)5 b Fr(\))26 b(=)f(det\()p Fj(A)p Fr(\).)41 b(Then)27 b Fj(R)3663 4839 y Ff(0)3715 4872 y Fr(is)g(no)0 4985 y(longer)k(a)g(subgroup)f(of)h Fi(SL)959 5004 y Fe(n)1002 4985 y Fh(2)8 b Fr(\()p Fj(F)14 b Fr(\),)32 b(but)e(its)h(in)m(tersection)g(with)f Fi(SL)2339 5004 y Fe(n)2382 4985 y Fh(2)8 b Fr(\()p Fj(F)14 b Fr(\))31 b(is)g(a)g(subgroup)f Fj(R)3228 4952 y Ff(00)3270 4985 y Fr(.)43 b(As)32 b(asserted)f(in)0 5098 y([MS02)q(],)j Fj(R)404 5065 y Ff(00)479 5098 y Fr(comprises)f(all)f(matrices)h(in)f Fi(SL)1611 5117 y Fe(n)1654 5098 y Fh(2)8 b Fr(\()p Fj(F)14 b Fr(\))33 b(that)h(\014x)e(det)q(,)i(so)f(the)g(isotrop)m(y)g (subgroup)f(of)h(det)g(under)0 5210 y(the)27 b(standard)e(action)i(b)m (y)g Fi(SL)1028 5230 y Fe(n)1071 5211 y Fh(2)1110 5210 y Fr(\()p Fj(F)13 b Fr(\))27 b(is)e Fj(R)1435 5177 y Ff(00)1478 5210 y Fr(.)39 b(Moreo)m(v)m(er,)30 b(the)c(only)g(p)s (olynomials)e(\014xed)i(b)m(y)g Fj(R)3222 5177 y Ff(00)3291 5210 y Fr(are)h(m)m(ultiples)d(of)0 5323 y(det,)34 b(so)f(in)e(pro)5 b(jectiv)m(e)34 b(space,)g(det)1229 5337 y Fe(n)1309 5323 y Fr(is)e(c)m(haracterized)i(b)m(y)e Fj(R)2157 5290 y Ff(00)2200 5323 y Fr(.)47 b(The)33 b(isotrop)m(y)f(subgroup)g(of)h (the)g(p)s(ermanen)m(t)p eop %%Page: 5 5 5 4 bop 0 91 a Fr(for)28 b Fj(n)d Fo(\025)g Fr(3)k(is)e(sho)m(wn)h (\([MS02)q(])h(citing)e(Minc\))i(to)g(b)s(e)e(generated)j(b)m(y)e (\(the)h(linear)e(transformations)g(in)h Fi(SL)3677 111 y Fe(n)3720 92 y Fh(2)3758 91 y Fr(\()p Fj(F)14 b Fr(\))0 204 y(arising)25 b(from\))i(the)g(sub)s(cases)g(where)f Fj(A)h Fr(and)g Fj(B)k Fr(are)d(either)e(diagonal)g(or)h(p)s(erm)m (utation)f(matrices,)i(pro)m(vided)e Fj(F)0 317 y Fr(is)32 b(not)h(of)g(c)m(haracteristic)g(2,)h(and)e(again)h(the)g(p)s(ermanen)m (t)f(is)g(the)h(unique)e(p)s(olynomial)f(up)i(to)h(m)m(ultiples)d(that) 0 430 y(it)g(\014xes.)141 543 y(Finally)-8 b(,)47 b(an)d(action)h(b)m (y)g(a)g(group)f Fj(G)g Fr(on)g Fj(V)65 b Fr(can)45 b(\014x)f(a)g(p)s (olynomial)e(function)i Fj(p)g Fr(on)g Fj(V)65 b Fr(in)43 b(the)i(sense)0 656 y(that)37 b(for)g(all)e Fj(g)k Fo(2)c Fj(G)p Fr(,)j Fj(\025x:p)p Fr(\()p Fj(\013)p Fr(\()p Fj(g)s(;)15 b(x)p Fr(\)\))40 b(equals)c Fj(p)g Fr(itself.)58 b(Then)36 b Fj(p)g Fr(is)g(lik)m(ewise)f(\\)p Fj(G)p Fr(-in)m(v)-5 b(arian)m(t.")59 b(When)36 b Fj(V)57 b Fr(is)36 b(a)0 769 y(space)g(of)g(p)s(olynomials,)e(thinking)f(of)j (\\p)s(olynomials)d(with)h(p)s(olynomial)f(argumen)m(ts")j(can)g(seem)f (hairy)-8 b(,)37 b(but)0 882 y(remem)m(b)s(ering)k(corresp)s(ondences)i (suc)m(h)f(as)h Fj(V)67 b Fr(=)42 b(the)h(span)g(of)g Fo(f)15 b Fj(x)2439 849 y Fm(3)2479 882 y Fj(;)g(x)2571 849 y Fm(2)2610 882 y Fj(y)s(;)g(xy)2798 849 y Fm(2)2838 882 y Fj(;)g(y)2926 849 y Fm(3)2981 882 y Fo(g)46 b Fr(=)g Fl(C)3265 849 y Fm(4)3347 882 y Fr(can)d(help.)77 b(As)0 995 y(ascrib)s(ed)26 b(to)i(Hilb)s(ert)d(in)i([MS02)q(],)h(the)g(n)m (ull)d(cone)j(of)g Fj(V)47 b Fr(under)26 b(a)i(giv)m(en)f(group)g (action)h(is)e(c)m(haracterized)j(as)f(the)0 1108 y(set)h(of)g(p)s(oin) m(ts)e(on)h(whic)m(h)f(ev)m(ery)j(non-constan)m(t)f(homogeneous)g(p)s (olynomial)d(that)j(is)e(constan)m(t)j(on)f(orbits)e(of)i Fj(G)0 1220 y Fr(v)-5 b(anishes.)47 b(This)31 b(is)i(an)f(example)h(of) g(ho)m(w)g(the)g(notion)g(of)g(isotrop)m(y)g(\(\\stabilizing"\))f(in)m (teracts)i(with)d(stabilit)m(y)0 1333 y(\(or)g(b)s(eing)e(unstable\).)0 1577 y Fd(2.2)112 b(Linear)38 b(group)g(represen)m(tations)0 1748 y Fr(An)22 b(action)h Fj(\013)464 1762 y Fe(G)546 1748 y Fr(on)f(a)h(v)m(ector)h(space)f Fj(V)42 b Fr(is)22 b Fi(line)-5 b(ar)33 b Fr(if)22 b(for)g(all)f Fj(g)29 b Fo(2)c Fj(G)p Fr(,)f Fj(u;)15 b(v)29 b Fo(2)24 b Fj(V)d Fr(,)j(and)e(scalars)g Fj(c)k Fo(2)e Fj(F)13 b Fr(,)25 b Fj(\013)p Fr(\()p Fj(g)s(;)15 b(cu)t Fr(+)t Fj(v)s Fr(\))28 b(=)0 1861 y Fj(c\013)p Fr(\()p Fj(g)s(;)15 b(u)p Fr(\))f(+)g Fj(\013)p Fr(\()p Fj(g)s(;)h(v)s Fr(\).)44 b(Then)27 b Fj(\013)p Fr(\()p Fj(g)s(;)15 b Fo(\001)p Fr(\))29 b(induces)d(a)i(linear)d(transformation)i(on)g Fj(V)20 b Fr(,)28 b(so)g(there)f(is)g(a)g(matrix)g Fj(A)3606 1875 y Fe(g)3673 1861 y Fr(of)h(the)0 1974 y(same)j(dimension)d(as)j Fj(V)50 b Fr(suc)m(h)30 b(that)h Fj(\013)p Fr(\()p Fj(g)s(;)15 b(u)p Fr(\))28 b(=)d Fj(A)1733 1988 y Fe(g)1773 1974 y Fj(u)31 b Fr(for)f(all)g Fj(u)25 b Fo(2)g Fj(V)20 b Fr(.)41 b(Th)m(us)30 b(the)g(action)h(of)g(left)f(m)m(ultiplication)0 2087 y(b)m(y)j(a)g(matrix)f(is)g(canonical,)h(and)f(one)h(can)h(regard) e Fj(\013)i Fr(itself)d(as)i(a)g(homomorphism)e(from)h Fj(G)h Fr(in)m(to)f(a)i(group)e(of)0 2200 y(suc)m(h)j(matrices.)57 b(In)35 b(the)h(action)g Fj(\013)p Fr(\()p Fj(B)5 b(;)15 b(h)p Fr(\))36 b(=)e Fj(\025x:h)p Fr(\()p Fj(B)1911 2167 y Ff(\000)p Fm(1)2006 2200 y Fj(x)p Fr(\))i(used)f(b)m(y)g(Mulm)m(uley) f(and)h(Sohoni)g(ab)s(o)m(v)m(e,)j(\\)p Fj(A)3793 2214 y Fe(B)3855 2200 y Fr(")0 2313 y(b)s(ecomes)31 b(an)f(exp)s(onen)m (tially)e(large)j(matrix)f(in)f(terms)h(of)h Fj(m)f Fr(\(if)f Fj(d)d Fr(=)f(\002\()p Fj(m)p Fr(\)\).)141 2426 y(An)i(action)g Fj(\013)614 2440 y Fe(H)708 2426 y Fr(b)m(y)g(a)h(homomorphic)d(image)j Fj(H)k Fr(=)25 b Fj(\036)p Fr(\()p Fj(G)p Fr(\))i(of)g Fj(G)g Fr(can)g(b)s(e)f(regarded)h(as)g(an)g(action)h(of)f Fj(G)f Fr(itself)0 2539 y(via)k Fj(\013)207 2553 y Fe(G)266 2539 y Fr(\()p Fj(g)s(;)15 b(u)p Fr(\))27 b(=)e Fj(\013)655 2553 y Fe(H)723 2539 y Fr(\()p Fj(\036)p Fr(\()p Fj(g)s Fr(\))p Fj(;)15 b(u)p Fr(\),)33 b(since)1205 2743 y Fj(\013)1263 2757 y Fe(G)1322 2743 y Fr(\()p Fj(g)s(h;)15 b(u)p Fr(\))85 b(=)e Fj(\013)1879 2757 y Fe(H)1947 2743 y Fr(\()p Fj(\036)p Fr(\()p Fj(g)s(h)p Fr(\))p Fj(;)15 b(u)p Fr(\))1667 2881 y(=)83 b Fj(\013)1879 2895 y Fe(H)1947 2881 y Fr(\()p Fj(\036)p Fr(\()p Fj(g)s Fr(\))p Fj(\036)p Fr(\()p Fj(h)p Fr(\))p Fj(;)15 b(u)p Fr(\))1667 3019 y(=)83 b Fj(\013)1879 3033 y Fe(H)1947 3019 y Fr(\()p Fj(\036)p Fr(\()p Fj(g)s Fr(\))p Fj(;)15 b(\013)2250 3033 y Fe(H)2319 3019 y Fr(\()p Fj(\036)p Fr(\()p Fj(h)p Fr(\))p Fj(;)g(u)p Fr(\)\))1667 3156 y(=)83 b Fj(\013)1879 3170 y Fe(G)1938 3156 y Fr(\()p Fj(g)s(;)15 b(\013)2117 3170 y Fe(G)2178 3156 y Fr(\()p Fj(h;)g(u)p Fr(\)\))0 3361 y(.)41 b(Th)m(us)29 b(if)g Fj(S)36 b Fr(is)29 b(an)h Fj(H)7 b Fr(-mo)s(dule,)29 b(it)h(is)g(a)h Fj(G)p Fr(-mo)s(dule)d(for)i(an)m(y)h Fj(G)f Fr(of)g(whic)m(h)f Fj(H)38 b Fr(is)29 b(a)i(subgroup.)141 3474 y(Eviden)m(tly)e(b)s(ecause)h(of)g(these)g(facts,)h(it)f(has)g(b)s (ecome)g(\\cultural")f(to)i(call)e(either)h(the)g(matrix)f(group)h(or)g Fj(V)0 3587 y Fr(itself)35 b(a)h Fi(r)-5 b(epr)g(esentation)46 b Fr(of)36 b Fj(G)p Fr(,)h(ev)m(en)g(if)e(the)h(mapping)e(\\really")i (represen)m(ts)g(only)f(a)i(small)d(image)i Fj(H)43 b Fr(of)36 b Fj(G)p Fr(.)0 3699 y(Another)30 b(transm)m(utation)g(is)f (that)h(a)g(group)g Fj(G)f Fr(of)h Fj(n)19 b Fo(\002)g Fj(n)29 b Fr(matrices)h(o)m(v)m(er)h(a)g(\014eld)d Fj(F)43 b Fr(is-a)30 b(subset)f(of)h(the)g(v)m(ector)0 3812 y(space)f Fj(F)312 3779 y Fe(n)355 3756 y Fh(2)394 3812 y Fr(,)g(and)f(one)g(can) h(attribute)f(to)h Fj(G)f Fr(prop)s(erties)f(suc)m(h)h(as)h(b)s(eing)e (Z-closed,)i(lo)s(cally)e(closed,)i(connected,)0 3925 y(and/or)h(compact)i(\(the)f(latter)g(t)m(w)m(o)g(with)e(reference)i (to)g(the)g(Euclidean)d(top)s(ology\).)141 4038 y(F)-8 b(or)28 b(instance,)g(the)f Fi(sp)-5 b(e)g(cial)31 b(line)-5 b(ar)31 b(gr)-5 b(oup)34 b(SL)1728 4052 y Fe(n)1775 4038 y Fr(\()p Fj(F)13 b Fr(\))28 b(of)f Fj(n)14 b Fo(\002)g Fj(n)25 b Fr(matrices)i Fj(A)h Fr(with)d(det)q(\()p Fj(A)p Fr(\))h(=)f(1)i(is)f(Z-closed|)0 4151 y(b)s(ecause)31 b(it)f(is)g(de\014ned)f(b)m(y)i(the)g(single)e(equation)i(det\()p Fj(x)1955 4165 y Fm(11)2030 4151 y Fj(;)15 b(:)g(:)g(:)i(;)e(x)2284 4165 y Fe(nn)2374 4151 y Fr(\))26 b(=)g(1.)42 b(It)31 b(is)f Fi(c)-5 b(onne)g(cte)g(d)42 b Fr(for)30 b Fj(F)39 b Fr(=)26 b Fl(Q)p Fj(;)15 b Fl(R)p Fj(;)g Fl(C)0 4264 y Fr(b)s(ecause)36 b(an)m(y)h(t)m(w)m(o)h(matrices)e(of)g(determinan)m (t)g(1)h(can)g(b)s(e)e(con)m(tin)m(uously)g(v)-5 b(aried)36 b(one)g(to)i(the)e(other)h(through)0 4377 y(matrices)27 b(of)g(determinan)m(t)f(1.)40 b(The)27 b Fi(gener)-5 b(al)30 b(line)-5 b(ar)30 b(gr)-5 b(oup)34 b(GL)2204 4391 y Fe(n)2251 4377 y Fr(\()p Fj(F)13 b Fr(\))27 b(of)g(in)m(v)m (ertible)f Fj(n)13 b Fo(\002)g Fj(n)26 b Fr(matrices)h(is)e(Z-op)s(en,) 0 4490 y(b)s(ecause)34 b(it)f(is)f(the)i(complemen)m(t)g(of)g(the)g (Z-closed)f(set)h(de\014ned)e(b)m(y)i(det\()p Fj(x)2627 4504 y Fm(11)2702 4490 y Fj(;)15 b(:)g(:)g(:)i(;)e(x)2956 4504 y Fe(nn)3046 4490 y Fr(\))31 b(=)f(0.)51 b(It)34 b(is,)f(ho)m(w)m(ev)m(er,)0 4603 y(lo)s(cally)c(Z-closed.)141 4716 y(What)24 b(matters)g(to)g(us)e(most)i(in)e(a)h(represen)m (tation,)i(ho)m(w)m(ev)m(er,)i(is)22 b(the)h(corresp)s(ondence)g(b)s (et)m(w)m(een)h(subgroups)0 4829 y Fj(H)37 b Fr(of)29 b Fj(G)g Fr(and)g(subspaces)g Fj(W)43 b Fr(of)29 b Fj(V)50 b Fr(that)30 b(are)g(stabilized)e(\(one)j(also)e(sa)m(ys)h Fi(pr)-5 b(eserve)g(d)41 b Fr(or)30 b Fi(invariant)9 b Fr(\))31 b(under)d(the)0 4941 y(action)23 b(of)f Fj(G)p Fr(.)37 b(If)22 b Fj(H)29 b Fr(stabilizes)21 b Fj(W)13 b Fr(,)24 b(then)e Fj(W)34 b Fr(is)21 b(a)i(represen)m(tation)f(of)h Fj(H)7 b Fr(.)38 b(A)22 b(represen)m(tation)g Fj(\013)3244 4955 y Fe(G)3326 4941 y Fr(is)f Fi(irr)-5 b(e)g(ducible)30 b Fr(if)0 5054 y(there)25 b(is)f(no)g(prop)s(er)f(subspace)h Fj(W)37 b Fr(of)25 b Fj(V)45 b Fr(that)25 b(is)f(stabilized)f(b)m(y)h Fj(G)p Fr(.)38 b(If)25 b Fj(G)f Fr(is)f(a)i(direct)f(sum)g Fj(H)16 b Fr(+)9 b Fj(J)33 b Fr(of)25 b(t)m(w)m(o)h(groups)0 5167 y Fj(H)32 b Fr(and)24 b Fj(J)33 b Fr(\(i.e.,)27 b Fj(G)d Fr(is)f(isomorphic)g(to)i Fj(H)16 b Fo(\002)9 b Fj(J)33 b Fr(with)24 b(the)g(group)g(pro)s(duct)g(\()p Fj(h)2588 5181 y Fm(1)2628 5167 y Fj(;)15 b(j)2705 5181 y Fm(1)2745 5167 y Fr(\))-6 b Fo(\001)g Fr(\()p Fj(h)2880 5181 y Fm(2)2920 5167 y Fj(;)15 b(j)2997 5181 y Fm(2)3037 5167 y Fr(\))26 b(=)f(\()p Fj(h)3281 5181 y Fm(1)3315 5167 y Fo(\001)-6 b Fj(h)3386 5181 y Fm(2)3425 5167 y Fj(;)15 b(j)3502 5181 y Fm(1)3536 5167 y Fo(\001)-6 b Fj(j)3592 5181 y Fm(2)3632 5167 y Fr(\)\),)27 b(and)0 5280 y Fj(V)45 b Fr(=)25 b Fj(W)k Fr(+)16 b Fj(X)36 b Fr(for)28 b(subspaces)g Fj(W)41 b Fr(and)28 b Fj(X)36 b Fr(preserv)m(ed)28 b(under)f(the)h(induced)f(actions)h Fj(\013)2985 5294 y Fe(H)3081 5280 y Fr(and)g Fj(\013)3314 5294 y Fe(J)3363 5280 y Fr(,)h(resp)s(ectiv)m(ely)-8 b(,)p eop %%Page: 6 6 6 5 bop 0 91 a Fr(then)33 b Fj(\013)268 105 y Fe(G)360 91 y Fr(is)f Fi(r)-5 b(e)g(ducible)41 b Fr(and)32 b Fi(factors)42 b Fr(as)34 b Fj(\013)1498 105 y Fe(G)1587 91 y Fr(=)29 b Fj(\013)1745 105 y Fe(H)1835 91 y Fr(+)21 b Fj(\013)1985 105 y Fe(J)2035 91 y Fr(.)48 b(W)-8 b(riting)33 b Fj(v)g Fo(2)c Fj(V)54 b Fr(uniquely)30 b(as)j Fj(w)25 b Fr(+)d Fj(x)33 b Fr(for)g Fj(w)f Fo(2)e Fj(W)0 204 y Fr(and)g Fj(x)25 b Fo(2)g Fj(X)7 b Fr(,)31 b(and)f Fj(g)e Fo(2)d Fj(G)30 b Fr(corresp)s(onds)f(to)i(\()p Fj(h;)15 b(j)5 b Fr(\))33 b(with)c Fj(h)c Fo(2)g Fj(H)37 b Fr(and)30 b Fj(j)h Fo(2)25 b Fj(J)9 b Fr(,)31 b(w)m(e)g(get)856 377 y Fj(\013)914 391 y Fe(G)973 377 y Fr(\()p Fj(g)s(;)15 b(u)p Fr(\))27 b(=)e(\()p Fj(h;)15 b(j)5 b Fr(\))g Fo(\001)g Fr(\()p Fj(w)25 b Fr(+)20 b Fj(x)p Fr(\))26 b(=)f Fj(hw)e Fr(+)d Fj(j)5 b(x)26 b Fr(=)f Fj(\013)2473 391 y Fe(H)2540 377 y Fr(\()p Fj(w)r Fr(\))d(+)d Fj(\013)2847 391 y Fe(J)2897 377 y Fr(\()p Fj(x)p Fr(\))p Fj(:)0 550 y Fr(Note)33 b(that)f(if)e Fj(G)h Fr(itself)f(stabilizes)g Fj(W)13 b Fr(,)31 b(then)g(w)m(e)h(can)g(tak)m(e)h Fj(H)h Fr(=)26 b Fj(G)31 b Fr(and)g(\\)p Fj(J)9 b Fr(")32 b(=)f(the)h(iden)m(tit)m(y)f (subgroup)e Fl(1)j Fr(of)0 662 y Fj(G)e Fr(and)g(factor)1556 775 y Fj(\013)1614 789 y Fe(G)1698 775 y Fr(=)25 b Fj(\013)1852 789 y Fe(G)1916 775 y Fo(o)5 b Fj(W)33 b Fr(+)20 b Fj(\034)15 b Fo(o)5 b Fj(X)r(;)0 926 y Fr(where)34 b Fj(\034)44 b Fr(is)34 b(the)g(action)h(of)f Fl(1)h Fr(and)f(is)f(called)h(the)g Fi(trivial)j(r)-5 b(epr)g(esentation)7 b Fr(.)56 b(\(Here)20 b Fo(o)f Fr(means)35 b(\\restricted)f(to."\))0 1039 y(Note)h(also)e (that)h Fj(\034)17 b Fo(o)7 b Fj(X)41 b Fr(itself)33 b(is)f(irreducible)e(if)j(and)g(only)f(if)g(the)i(subspace)f Fj(X)41 b Fr(is)32 b(1-dimensional.)48 b(The)33 b(main)0 1152 y(theorem)41 b(this)e(leads)g(to)i(is)e(that)i Fi(every)g(r)-5 b(epr)g(esentation)45 b(onto)d(a)g(\014nite-dimensional)i(ve)-5 b(ctor)42 b(sp)-5 b(ac)g(e)43 b(c)-5 b(an)42 b(b)-5 b(e)0 1265 y(factor)g(e)g(d)37 b(into)e(irr)-5 b(e)g(ducible)36 b(r)-5 b(epr)g(esentations)7 b Fr(,)37 b(and)32 b(this)f(factorization) j(is)e(unique)e(up)i(to)h(isomorphism.)45 b(The)0 1378 y(n)m(um)m(b)s(er)d(of)i(times)f Fj(\034)54 b Fr(o)s(ccurs)43 b(in)f(this)h(represen)m(tation)g(giv)m(es)h(information)e(ab)s(out)h (the)h(dimensionalit)m(y)d(of)0 1491 y(in)m(v)-5 b(arian)m(t)30 b(subspaces.)141 1604 y(Represen)m(tations)41 b(ha)m(v)m(e)h(immediate) e(relev)-5 b(ance)41 b(via)f Fi(Kempf)9 b('s)43 b(stability)g (criterion)7 b Fr(.)72 b(A)41 b Fi(one-p)-5 b(ar)g(ameter)0 1717 y(sub)g(gr)g(oup)42 b Fj(G)453 1684 y Ff(0)511 1717 y Fr(of)35 b Fj(G)g Fr(is)f(the)h(image)g(of)h(a)f(homomorphism)e(from) i(the)g(m)m(ultiplicativ)m(e)e(group)i Fj(F)3379 1681 y Ff(\003)3453 1717 y Fr(of)g(non-zero)0 1830 y(elemen)m(ts)h(of)f Fj(F)48 b Fr(to)36 b Fj(G)p Fr(.)54 b(It)36 b(is)e Fi(c)-5 b(entr)g(al)46 b Fr(if)34 b(ev)m(ery)i(elemen)m(t)g(of)f Fj(G)2220 1797 y Ff(0)2278 1830 y Fr(comm)m(utes)h(with)e(ev)m(ery)i (elemen)m(t)f(of)h Fj(G)p Fr(.)54 b(F)-8 b(or)0 1943 y(example,)30 b Fi(GL)509 1957 y Fe(n)556 1943 y Fr(\()p Fj(F)13 b Fr(\))30 b(has)g(the)h(non)m(trivial)d(cen)m(tral)j (one-parameter)g(subgroup)e(consisting)g(of)h(scalar)g(m)m(ultiples)0 2056 y(of)35 b(the)h(iden)m(tit)m(y)f(matrix,)h(but)e Fi(SL)1217 2070 y Fe(n)1264 2056 y Fr(\()p Fj(F)14 b Fr(\))35 b(do)s(es)g(not|and)g(has)g(no)g(suc)m(h)g(subgroup.)53 b(F)-8 b(ollo)m(wing)35 b(Alp)s(erin)e(and)0 2168 y(Bell)41 b([AB95)q(],)k(de\014ne)c(a)g(subgroup)f Fj(G)1378 2136 y Ff(0)1442 2168 y Fr(of)i Fj(G)f Fr(to)h(b)s(e)f Fi(p)-5 b(ar)g(ab)g(olic)49 b Fr(if)40 b(it)h(is)g(the)g(sim)m(ultaneous)f (stabilizer)g(of)i(a)0 2281 y(sequence)31 b(of)f(linear)f(subspaces) 1175 2454 y(0)c Fo(\032)g Fj(W)1427 2468 y Fm(1)1492 2454 y Fo(\032)g Fj(W)1674 2468 y Fm(2)1738 2454 y Fo(\032)g Fj(:)15 b(:)g(:)27 b Fo(\032)e Fj(W)2148 2468 y Fe(r)r Ff(\000)p Fm(1)2301 2454 y Fo(\032)g Fj(W)2483 2468 y Fe(r)2546 2454 y Fr(=)g Fj(V)5 b(:)0 2627 y Fr(Suc)m(h)24 b(a)h(sequence)g(is)f(called)g(a)h Fi(\015ag)8 b Fr(,)26 b(and)e(b)m(y)h(the)g(prop)s(er)e(con)m(tainmen)m(ts,)k(m)m(ust)d(ha)m (v)m(e)i Fj(r)i Fo(\024)d Fj(n)g Fr(=)g(dim)n(\()p Fj(V)c Fr(\))k(terms.)0 2740 y(F)-8 b(or)23 b(groups)e Fj(G)h Fr(suc)m(h)g(as)g Fi(SL)942 2754 y Fe(n)989 2740 y Fr(\()p Fj(F)13 b Fr(\))23 b(with)e(no)h(non)m(trivial)e(cen)m(tral)i (one-parameter)i(subgroups,)e(Kempf)7 b('s)21 b(criterion)0 2853 y(states)36 b(that)f(a)g(pro)5 b(jectiv)m(e)35 b(p)s(oin)m(t)f Fj(y)h Fo(2)d Fj(P)13 b Fr(\()p Fj(V)21 b Fr(\))35 b(is)f(stable)g(if)g (the)h(isotrop)m(y)f(subgroup)f Fj(G)3071 2867 y Fe(y)3147 2853 y Fr(is)g(not)i(con)m(tained)g(in)0 2965 y(an)m(y)d(prop)s(er)e (parab)s(olic)g(subgroup)g(of)i Fj(G)p Fr(.)44 b(In)31 b(particular)f(this)g(follo)m(ws)h(if)g(the)h(represen)m(tation)f(of)h Fj(G)3534 2979 y Fe(y)3607 2965 y Fr(on)f Fj(V)52 b Fr(is)0 3078 y(irreducible.)k(The)36 b(ab)s(o)m(v)m(e)h(descriptions)e(of)h (the)h(isotrop)m(y)f(subgroups)f(of)i(det)2768 3092 y Fe(n)2851 3078 y Fr(and)f(p)s(erm)3239 3100 y Fe(n)3322 3078 y Fr(are)h(irreducible)0 3191 y(represen)m(tations,)31 b(so)f(these)h(p)s(olynomials)d(are)i(stable)g(under)f(the)i(action)g (b)m(y)f Fi(SL)2842 3211 y Fe(n)2885 3192 y Fh(2)2924 3191 y Fr(\()p Fj(F)13 b Fr(\).)0 3473 y Fk(3)135 b(Application)45 b(to)g(Complexit)l(y)i(Theory)0 3676 y Fr(The)g(jumping-o\013)f(p)s (oin)m(t)g(for)h(the)g(Mulm)m(uley-Sohoni)e(metho)s(d)i(is)f(V)-8 b(alian)m(t's)48 b(metho)s(d)f([V)-8 b(al79)q(])48 b(\(see)g(also)0 3789 y([vzG87)r(]\))27 b(of)h(reducing)d(an)m(y)j(p)s(olynomial-size)c (family)i(of)h(arithmetical)f(circuits)g(to)h(a)h(p)s(olynomial-size)d (family)0 3902 y(of)f(determinan)m(t)g(computations.)38 b(This)22 b(extends)i(to)h(sa)m(ying)e(that)i(functions)d(b)s(eliev)m (ed)h(to)i(b)s(e)e(in)m(tractable,)i(suc)m(h)0 4014 y(as)35 b(the)g(p)s(ermanen)m(t)f(p)s(olynomials,)f(ha)m(v)m(e)j(p)s (olynomial-size)d(\(arithmetical\))h(circuits)g(i\013)g(they)g(b)s (elong)g(to)i(the)0 4127 y(\(Z-closure)c(of)g(the\))g(orbits)f(of)h (the)g(determinan)m(t)f(p)s(olynomials)f(under)g(certain)i(group)f (actions.)46 b(It)32 b(is)f(imp)s(or-)0 4240 y(tan)m(t)h(to)f(note)g (that)g(these)h(p)s(olynomials)27 b(are)k(represen)m(ted)g(as)g(v)m (ectors)h(of)f(length)f(exp)s(onen)m(tial)f(in)g(\\)p Fj(n)p Fr(,")j(hence)0 4353 y(exp)s(onen)m(tial)e(in)f(the)i(size)g(of) g(the)g(matrices)g(in)m(v)m(olv)m(ed)g(at)g(the)g(outset.)43 b(W)-8 b(e)32 b(henceforth)f(alter)g(the)g(notation)g(in)0 4466 y([MS02)q(])g(to)g(mak)m(e)g(explicit)e(a)i(distinction)d(b)s(et)m (w)m(een)j Fi(lar)-5 b(ge)38 b Fr(and)29 b Fi(smal)5 b(l)41 b Fr(ob)5 b(jects.)111 4647 y(1.)46 b Fj(n)30 b Fr(is)g(alw)m(a)m(ys)g(the)h(reference)g(parameter)g(for)f(the)g (length)g(of)h(the)f(input)f(to)i(a)g(computational)f(problem.)111 4824 y(2.)46 b Fj(N)35 b Fr(=)25 b Fj(n)486 4791 y Fm(2)547 4824 y Fr(is)c(the)g(size)h(of)g(an)f Fj(n)s Fo(\002)s Fj(n)f Fr(matrix.)37 b(Since)21 b(w)m(e)h(are)g(concerned)g(only)e (with)h(p)s(olynomial)e(complexit)m(y)-8 b(,)227 4937 y(w)m(e)31 b(can)g(measure)f(in)f(terms)h(of)h Fj(n)f Fr(ev)m(en)h(if)e(the)i(input)d(is)i(a)g(matrix)g(of)h(size)f Fj(N)10 b Fr(.)111 5114 y(3.)46 b Fj(m)d Fr(=)f Fj(n)518 5081 y Fe(O)r Fm(\(1\))708 5114 y Fr(is)d(the)i(n)m(um)m(b)s(er)f(of)g (ro)m(ws/columns)g(in)g(paddings)e(of)j Fj(n)27 b Fo(\002)f Fj(n)41 b Fr(matrices)f(that)i(arise,)h(and)227 5227 y(b)s(ecomes)i(the)g(degree)g(of)f(the)h(p)s(ermanen)m(t,)i(determinan) m(t,)h(and)c(other)h(homogeneous)g(p)s(olynomials)227 5340 y(asso)s(ciated)31 b(to)g(these)g(matrices.)41 b(Sometimes)30 b Fj(d)g Fr(stands)g(for)g(the)h(degree)g(of)g(these)f(p)s(olynomials.) p eop %%Page: 7 7 7 6 bop 111 91 a Fr(4.)46 b Fj(M)36 b Fr(=)25 b Fj(m)527 58 y Fm(2)596 91 y Fr(is)30 b(the)g(size)g(of)h(this)e(matrix,)h(and)g (also)g(the)h(n)m(um)m(b)s(er)e(of)h(v)-5 b(ariables)29 b(of)i(these)g(p)s(olynomials.)111 274 y(5.)46 b Fj(A;)15 b(B)5 b(;)15 b(C)q(;)g(:)g(:)g(:)33 b Fr(stand)d(for)g Fj(n)20 b Fo(\002)g Fj(n)30 b Fr(or)g Fj(m)20 b Fo(\002)g Fj(m)30 b Fr(matrices.)111 456 y(6.)46 b Fj(p)p Fr(\()p Fj(n)p Fr(\))26 b(=)e Fj(n)574 423 y Fe(O)r Fm(\(1\))754 456 y Fr(stands)30 b(for)g(a)h(p)s(olynomial)d(running)f(time.)111 638 y(7.)46 b Fj(f)5 b(;)15 b(g)s(;)g(h;)g(:)g(:)g(:)36 b Fr(stand)c(for)h(p)s(olynomials)d(in)h Fj(N)43 b Fr(or)32 b Fj(M)43 b Fr(v)-5 b(ariables|note)32 b(that)i(these)f(are)g(v)m (ectors)i(of)e(length)227 751 y(exp)s(onen)m(tial)d(in)f Fj(m)p Fr(,)h(i.e.)h Fi(lar)-5 b(ge)38 b Fr(ob)5 b(jects.)111 933 y(8.)46 b Fj(V)5 b(;)15 b(W)m(;)g(:)g(:)g(:)32 b Fr(stand)d(for)h(v)m(ector)h(spaces)f(of)g Fi(smal)5 b(l)40 b Fr(ob)5 b(jects,)30 b(while)e Fo(V)7 b Fj(;)15 b Fo(W)37 b Fr(stand)30 b(for)f(v)m(ector)i(spaces)f(of)g Fi(lar)-5 b(ge)227 1046 y Fr(ob)5 b(jects.)111 1229 y(9.)46 b Fj(G;)15 b(H)r(;)g(K)q(;)g(:)g(:)g(:)33 b Fr(stand)d(for)g(groups)g (of)g(small)f(matrices.)0 1470 y Fd(3.1)112 b(P)m(ermanen)m(t)37 b(and)h(Determinan)m(t)0 1641 y Fr(Fix)26 b Fj(F)40 b Fr(to)28 b(b)s(e)e(the)h(complex)f(n)m(um)m(b)s(ers)g(for)g(the)h(time) g(b)s(eing.)38 b(Supp)s(ose)25 b(the)i(p)s(ermanen)m(t)f(p)s(olynomial) e(p)s(erm)3753 1663 y Fe(n)3827 1641 y Fr(of)0 1754 y(an)i Fj(n)12 b Fo(\002)g Fj(n)26 b Fr(matrix)g Fj(x)697 1768 y Fe(ij)784 1754 y Fr(of)g(indeterminates)f(has)i(arithmetical)e(form)m (ulas)h(o)m(v)m(er)h Fj(F)40 b Fr(of)26 b(size)h Fj(s)p Fr(.)39 b(Then)25 b(as)i(remark)m(ed)0 1867 y(b)m(y)e(v)m(on)h(zur)f (Gathen)h([vzG87)r(],)h(there)e(is)g(an)g(\()p Fj(s)10 b Fr(+)g(2\))g Fo(\002)g Fr(\()p Fj(s)g Fr(+)g(2\))28 b(matrix)d Fj(M)36 b Fr(whose)25 b(en)m(tries)g(are)h(either)f(constan) m(ts)0 1980 y(or)35 b(v)-5 b(ariables)34 b Fj(x)550 1994 y Fe(ij)610 1980 y Fr(,)i(suc)m(h)f(that)h(det\()p Fj(M)10 b Fr(\))36 b(re-creates)g(the)f(p)s(ermanen)m(t)g(p)s(olynomial.)52 b(Th)m(us)34 b(p)s(erm)3406 2002 y Fe(n)3488 1980 y Fr(b)s(ecomes)i(a)0 2093 y(\\V)-8 b(alian)m(t)27 b(pro)5 b(jection")27 b(of)f(the)h (determinan)m(t)f(p)s(olynomial)e(of)i(an)h Fj(m)12 b Fo(\002)g Fj(m)26 b Fr(matrix)g(of)h(indeterminates)e Fj(y)3611 2107 y Fe(ij)3671 2093 y Fr(,)i(with)0 2206 y Fj(m)h Fr(=)f Fj(s)21 b Fr(+)g(2.)45 b(In)32 b(order)f(to)i(apply)d (the)i(Mulm)m(uley-Sohoni)d(setup,)k(w)m(e)f(need)f(to)i(pad)e(p)s(erm) 3195 2228 y Fe(n)3274 2206 y Fr(in)m(to)h(a)g(degree-)p Fj(m)0 2319 y Fr(homogeneous)41 b(p)s(olynomial)d(o)m(v)m(er)j(the)g Fj(y)1462 2333 y Fe(ij)1562 2319 y Fr(v)-5 b(ariables)39 b(of)h(equiv)-5 b(alen)m(t)40 b(circuit)f(complexit)m(y)-8 b(.)71 b(Mulm)m(uley)39 b(and)0 2432 y(Sohoni)h(do)h(this)g(via)g(a)h (map)f Fj(\036)p Fr(\()p Fj(f)10 b Fr(\))44 b(=)f Fj(y)1480 2399 y Fe(m)p Ff(\000)p Fe(n)1477 2454 y(mm)1644 2432 y Fj(f)1699 2399 y Ff(0)1722 2432 y Fr(,)h(where)d Fj(f)2120 2399 y Ff(0)2184 2432 y Fr(is)g Fj(f)50 b Fr(with)40 b(v)-5 b(ariables)41 b Fj(x)3041 2446 y Fe(ij)3142 2432 y Fr(renamed)g(to)i Fj(y)3690 2446 y Fe(ij)3791 2432 y Fr(for)0 2545 y(1)26 b Fo(\024)f Fj(i;)15 b(j)31 b Fo(\024)25 b Fj(n)30 b Fr(and)f(the)h(c)m(hoice)h(of)f Fj(y)1237 2559 y Fe(mm)1395 2545 y Fr(outside)g(this)e(square)i(b)s (eing)f(arbitrary)-8 b(.)39 b(W)-8 b(rite)31 b Fj(\036)p Fr(\()p Fj(f)10 b Fr(\))30 b(as)g Fj(f)3458 2512 y Fe(\036)3533 2545 y Fr(for)g(short.)0 2657 y(Also)c(let)f Fj(s)370 2672 y Fm(det)472 2657 y Fr(\()p Fj(n)p Fr(\))h(stand)g(for)f(an)h(upp) s(er)e(b)s(ound)f(on)j(the)g(form)m(ula)f(size)h(of)g(det)2650 2671 y Fe(n)2697 2657 y Fr(;)i(curren)m(tly)d Fj(s)3178 2676 y Fm(det\()p Fe(n)p Fm(\))3402 2657 y Fr(=)g Fj(n)3553 2624 y Fe(O)r Fm(\(log)13 b Fe(n)p Fm(\))3839 2657 y Fr(is)0 2770 y(b)s(est)25 b(kno)m(wn)f(\(see)i([BCS97)q(]\).)39 b(The)25 b(connection)g(to)h(complexit)m(y)e(theory)i(is)e(the)h(follo) m(wing)e(prop)s(osition)g(and)i(its)0 2883 y(near-con)m(v)m(erse,)33 b(whic)m(h)c(hold)g(for)i(an)m(y)f Fj(f)40 b Fr(in)30 b(place)g(of)h(p)s(erm)2096 2905 y Fe(n)2143 2883 y Fr(.)41 b(Note)32 b(that)f(unlik)m(e)e(the)i(de\014nition)d(of)j(stabilit)m(y) -8 b(,)0 2996 y(it)30 b(talks)g(ab)s(out)g(orbits)g(under)e(actions)j (in)e(pro)5 b(jectiv)m(e)31 b(space)g(itself.)0 3193 y Fl(Prop)s(osition)36 b(3.1)g(\([MS02],)f(Prop)s(ositions)i(4.1)e(and) g(4.4\))46 b Fi(With)97 b(r)-5 b(efer)g(enc)g(e)98 b(to)g(the)g(action) g(of)0 3306 y(SL)108 3325 y Fe(m)170 3306 y Fh(2)9 b Fr(\()p Fj(F)14 b Fr(\))33 b Fi(on)g(p)-5 b(olynomials)36 b(in)c Fj(m)1200 3273 y Fm(2)1272 3306 y Fi(variables:)61 3480 y(\(a\))46 b(If)37 b(the)g(p)-5 b(ermanent)39 b(has)f(formulas)g (of)f(size)g Fj(m)23 b Fo(\000)g Fr(2)p Fi(,)38 b(then)f Fr(p)s(erm)2493 3447 y Fe(\036)2493 3502 y(n)2576 3480 y Fi(is)g(in)g(the)g(pr)-5 b(oje)g(ctive)38 b(closur)-5 b(e)37 b(of)h(the)227 3593 y(orbit)33 b(of)g Fr(det)678 3607 y Fe(m)744 3593 y Fi(.)66 3775 y(\(b\))45 b(If)37 b Fr(p)s(erm)532 3742 y Fe(\036)532 3798 y(n)615 3775 y Fi(is)g(in)g(the)g(pr)-5 b(oje)g(ctive)38 b(closur)-5 b(e)37 b(of)g(the)g(orbit)h(of)f Fr(det)2434 3789 y Fe(m)2501 3775 y Fi(,)g(then)g(for)h(any)f Fj(\017)c(>)f Fr(0)p Fi(,)38 b(ther)-5 b(e)38 b(ar)-5 b(e)37 b Fj(n)3828 3742 y Fm(2)3867 3775 y Fi(-)227 3888 y(variable)d(formulas)h Fj(F)1002 3902 y Fe(\017)1068 3888 y Fi(of)f(size)f Fj(m)1433 3855 y Fm(2)1472 3888 y Fj(s)1515 3903 y Fm(det)1617 3888 y Fr(\()p Fj(m)p Fr(\))h Fi(that)g(appr)-5 b(oximate)37 b Fr(p)s(erm)2707 3910 y Fe(n)2754 3888 y Fi(,)d(in)f(the)g(str)-5 b(ong)35 b(sense)e(that)i(the)227 4001 y(c)-5 b(o)g(e\016cients)34 b(of)f(the)g(p)-5 b(olynomial)35 b(c)-5 b(ompute)g(d)35 b(by)e Fj(F)1983 4015 y Fe(\017)2048 4001 y Fi(ar)-5 b(e)34 b(within)f Fj(\017)g Fi(of)g(those)g(of)g(c)-5 b(orr)g(esp)g(onding)36 b(terms)e(of)227 4114 y Fr(p)s(erm)432 4136 y Fe(n)479 4114 y Fi(.)42 b(Only)32 b(the)h(c)-5 b(onstants)35 b(in)e Fj(F)1490 4128 y Fe(\017)1555 4114 y Fi(dep)-5 b(end)34 b(on)f Fj(\017)p Fi(.)0 4310 y Fl(Pro)s(of.)46 b Fr(\(a\))38 b(F)-8 b(orm)38 b(an)e Fj(m)25 b Fo(\002)f Fj(m)36 b Fr(matrix)g Fj(M)1575 4278 y Ff(0)1636 4310 y Fr(b)m(y)g(taking)h Fj(M)47 b Fr(ab)s(o)m(v)m(e,)40 b(m)m(ultiplying)33 b(ev)m(ery)38 b(nonzero)f(constan)m(t)0 4423 y(en)m(try)25 b(of)g Fj(M)35 b Fr(b)m(y)25 b Fj(y)619 4437 y Fe(mm)748 4423 y Fr(,)h(and)e(renaming)g(en)m(tries)h Fj(x)1702 4437 y Fe(ij)1787 4423 y Fr(with)f(1)h Fo(\024)g Fj(i;)15 b(j)32 b Fo(\024)25 b Fj(n)f Fr(bac)m(k)i(to)g Fj(y)2828 4437 y Fe(ij)2888 4423 y Fr(.)38 b(Then)24 b(det)q(\()p Fj(M)3443 4390 y Ff(0)3466 4423 y Fr(\))i(=)f(p)s(erm)3828 4390 y Fe(\036)3828 4446 y(n)3875 4423 y Fr(.)0 4536 y(Since)33 b(ev)m(ery)i(en)m(try)f(of)g Fj(M)931 4503 y Ff(0)988 4536 y Fr(is)f(a)i(trivial)d(linear)g(com)m(bination)i(of)g (en)m(tries)g(of)g(the)g Fj(m)22 b Fo(\002)h Fj(m)33 b Fr(matrix)h Fj(Y)51 b Fr(=)31 b(\()p Fj(y)3779 4550 y Fe(ij)3839 4536 y Fr(\),)0 4663 y(there)23 b(is)f(a)i(linear)d (transformation)h Fj(A)k Fr(:)f Fj(F)1450 4627 y Fe(m)1512 4603 y Fh(2)1577 4663 y Fo(!)g Fj(F)1764 4627 y Fe(m)1826 4603 y Fh(2)1888 4663 y Fr(that)f(pro)s(duces)e Fj(M)2551 4630 y Ff(0)2597 4663 y Fr(from)g Fj(Y)e Fr(.)39 b(Th)m(us)22 b(p)s(erm)3368 4630 y Fe(\036)3368 4685 y(n)3441 4663 y Fr(=)j(det\()p Fj(AY)20 b Fr(\).)0 4775 y(Although)29 b Fj(A)g Fr(ma)m(y)h(b)s(e)f(a)h(singular)d Fj(m)1315 4742 y Fm(2)1373 4775 y Fo(\002)18 b Fj(m)1542 4742 y Fm(2)1611 4775 y Fr(matrix,)29 b(there)h(are)f(elemen)m(ts)h Fj(A)2752 4742 y Ff(0)2805 4775 y Fr(of)g Fi(GL)3035 4795 y Fe(m)3097 4776 y Fh(2)3166 4775 y Fr(arbitrarily)c(close)k(to)0 4888 y Fj(A)p Fr(.)40 b(In)28 b(fact,)i(det\()p Fj(A)677 4855 y Ff(0)701 4888 y Fj(Y)20 b Fr(\))29 b(can)f(b)s(e)g(made)h(to)g (appro)m(ximate)f(p)s(erm)2202 4855 y Fe(\036)2202 4911 y(n)2278 4888 y Fr(co)s(e\016cien)m(t-wise)g(in)f(the)i(sense)f(of)h (\(b\).)40 b(\(Note)0 5001 y(that)d Fj(Y)57 b Fr(is)35 b(\\unrolled")g(as)i(a)g(v)m(ector)h(of)f(length)f Fj(m)1814 4968 y Fm(2)1889 5001 y Fr(in)g(the)g(pro)s(duct)g Fj(A)2580 4968 y Ff(0)2603 5001 y Fj(Y)20 b Fr(,)38 b(not)f(k)m(ept)g(as)g(a)g (matrix.\))59 b(Th)m(us)0 5114 y(p)s(erm)205 5081 y Fe(\036)205 5137 y(n)282 5114 y Fr(is)29 b(in)g(the)i(classical)e(closure)h(of)g (the)h Fi(GL)1676 5134 y Fe(m)1738 5115 y Fh(2)1777 5114 y Fr(-orbit)f(of)g(det)2259 5128 y Fe(m)2351 5114 y Fr(=)25 b(det)q(\()p Fj(Y)20 b Fr(\))30 b(in)f(a\016ne)i(space)f Fo(V)3394 5128 y Fe(m)3461 5114 y Fr(.)40 b(No)m(w)31 b(tak)m(e)0 5227 y Fj(d)d Fr(=)g(det\()p Fj(A)403 5194 y Ff(0)427 5227 y Fr(\))33 b(and)e Fj(A)741 5194 y Ff(00)812 5227 y Fr(=)c Fj(A)978 5194 y Ff(0)1002 5227 y Fj(=)p Fr(\()p Fj(d)1129 5194 y Fm(1)p Fe(=m)1261 5171 y Fh(2)1302 5227 y Fr(\).)46 b(Then)31 b(det\()p Fj(A)1876 5194 y Ff(00)1919 5227 y Fr(\))d(=)g(1,)33 b(so)f Fj(A)2365 5194 y Ff(00)2436 5227 y Fo(2)c Fi(SL)2633 5247 y Fe(m)2695 5228 y Fh(2)2734 5227 y Fr(,)33 b(and)e(det\()p Fj(A)3199 5194 y Ff(00)3242 5227 y Fj(Y)20 b Fr(\))33 b(is)e(co)s(e\016cien)m(t-) 0 5340 y(wise)h(a)h(scalar)f(m)m(ultiple)e(of)j(the)g(previous)e(appro) m(ximation)g(to)j(p)s(erm)2452 5307 y Fe(\036)2452 5362 y(n)2499 5340 y Fr(.)47 b(Th)m(us)31 b(passing)h(to)h(pro)5 b(jectiv)m(e)33 b(space,)p eop %%Page: 8 8 8 7 bop 0 91 a Fr(p)s(erm)205 58 y Fe(\036)205 114 y(n)293 91 y Fr(b)s(elongs)39 b(to)j(the)f(classical)f(closure)g(of)h(the)g Fi(SL)1981 111 y Fe(m)2043 92 y Fh(2)2083 91 y Fr(-orbit)f(of)h(det) 2586 105 y Fe(m)2653 91 y Fr(.)72 b(Since)40 b(the)h(classical)f (closure)g(is)0 204 y(alw)m(a)m(ys)31 b(con)m(tained)g(in)e(the)h (Z-closure,)g(\(a\))i(is)d(pro)m(v)m(ed.)141 317 y(\(b\))38 b(Supp)s(ose)d(p)s(erm)868 284 y Fe(\036)868 340 y(n)915 317 y Fr(,)k(regarded)e(as)h(a)f(p)s(oin)m(t)g(in)f Fj(P)13 b Fr(\()p Fo(V)2081 331 y Fe(m)2147 317 y Fr(\),)40 b(lies)c(in)g(the)h (Z-closure)g(of)g(the)h Fi(SL)3466 337 y Fe(m)3528 318 y Fh(2)3567 317 y Fr(-orbit)f(of)0 430 y(det)126 444 y Fe(m)193 430 y Fr(.)51 b(Here)34 b(w)m(e)h(use)e(the)h(\\k)m(ey)h (facts")g(that)g(this)e(orbit)g(is)g(a)h(lo)s(cally)e(Z-closed)i (subset)f(of)h(a)g(Z-closed)g(set)g(in)0 543 y Fj(P)13 b Fr(\()p Fo(V)162 557 y Fe(m)229 543 y Fr(\))27 b(\(see)i([Hum81]\),)g (and)e(th)m(us)g(its)f(Z-closure)h(coincides)f(with)g(its)h(classical)f (closure)h(in)f Fj(P)13 b Fr(\()p Fo(V)3397 557 y Fe(m)3464 543 y Fr(\).)40 b(Th)m(us)26 b(for)0 656 y(an)m(y)h Fj(\016)i(>)c Fr(0,)j(w)m(e)g(can)f(\014nd)e(an)i Fj(m)1106 623 y Fm(2)1159 656 y Fo(\002)13 b Fj(m)1323 623 y Fm(2)1389 656 y Fr(matrix)26 b Fj(A)1749 671 y Fe(\016)1814 656 y Fr(with)f(det\()p Fj(A)2246 671 y Fe(\016)2285 656 y Fr(\))h(=)e(1)k(suc)m(h)e(that)i (det\()p Fj(A)3138 671 y Fe(\016)3176 656 y Fj(Y)20 b Fr(\))28 b(has)e(co)s(e\016cien)m(ts)0 769 y(within)36 b Fj(\016)42 b Fr(of)c(some)h(m)m(ultiple)d Fj(c)j Fr(of)f(p)s(erm)1478 736 y Fe(\036)1478 791 y(n)1525 769 y Fr(.)65 b(No)m(w)39 b(let)f Fj(A)2037 736 y Ff(0)2037 795 y Fe(\016)2114 769 y Fr(=)g Fj(A)2291 784 y Fe(\016)2329 769 y Fj(=c)2413 736 y Fm(1)p Fe(=m)2551 769 y Fr(.)64 b(Then)38 b(det\()p Fj(A)3115 736 y Ff(0)3115 795 y Fe(\016)3153 769 y Fj(Y)20 b Fr(\))39 b(has)f(co)s(e\016cien)m(ts)0 882 y(within)28 b Fj(\016)s(=c)k Fr(of)e(p)s(erm)750 849 y Fe(\036)750 904 y(n)797 882 y Fr(.)41 b(Pro)m(vided)29 b Fj(\016)g(<)c(\017=c)p Fr(,)31 b(det)q(\()p Fj(A)1820 849 y Ff(0)1820 908 y Fe(\016)1858 882 y Fj(Y)20 b Fr(\))31 b(has)f(co)s(e\016cien)m(ts)h (within)d Fj(\017)i Fr(of)g(p)s(erm)3277 849 y Fe(\036)3277 904 y(n)3324 882 y Fr(.)141 995 y(No)m(w)i(in)e(forming)f(det)q(\()p Fj(A)1022 962 y Ff(0)1022 1021 y Fe(\016)1060 995 y Fj(Y)20 b Fr(\),)32 b(w)m(e)f(can)h(zero)g(out)f(v)-5 b(ariables)30 b(that)h(do)g(not)h(o)s(ccur)f(in)e(p)s(erm)3304 962 y Fe(\036)3304 1017 y(n)3351 995 y Fr(,)i(as)h(the)f(terms)0 1108 y(in)m(v)m(olving)26 b(these)j(v)-5 b(ariables)26 b(m)m(ust)i(ha)m(v)m(e)h(co)s(e\016cien)m(ts)f(of)g(magnitude)f(less)g (than)h Fj(\017)f Fr(an)m(yw)m(a)m(y)-8 b(.)42 b(Also)27 b(set)i Fj(y)3605 1122 y Fe(mm)3759 1108 y Fr(=)c(1)0 1220 y(and)34 b(rename)h(the)f(remaining)f(v)-5 b(ariables)33 b Fj(y)1519 1234 y Fe(ij)1614 1220 y Fr(to)i Fj(x)1781 1234 y Fe(ij)1842 1220 y Fr(.)53 b(Then)34 b(ev)m(ery)h(en)m(try)g(of)g (the)g(length-)p Fj(m)3275 1187 y Fm(2)3348 1220 y Fr(v)m(ector)h Fj(A)3693 1187 y Ff(0)3693 1247 y Fe(\016)3731 1220 y Fj(Y)55 b Fr(is)0 1333 y(a)33 b(linear)f(com)m(bination)g(of)i(v)-5 b(ariables)31 b Fj(x)1392 1347 y Fe(ij)1486 1333 y Fr(plus)g(p)s (ossibly)f(a)k(constan)m(t)g(term,)g(where)f(the)g(co)s(e\016cien)m(ts) h(dep)s(end)0 1446 y(on)e Fj(\017)p Fr(.)45 b(When)32 b(w)m(e)g(roll)f Fj(A)869 1413 y Ff(0)869 1473 y Fe(\016)907 1446 y Fj(Y)52 b Fr(bac)m(k)32 b(in)m(to)g(a)g(matrix)g(and)f(comp)s (ose)h(this)f(with)g(form)m(ulas)g Fj( )k Fr(of)d(size)g Fj(s)3507 1461 y Fm(det)3609 1446 y Fr(\()p Fj(m)p Fr(\))g(for)0 1559 y(the)h Fj(m)22 b Fo(\002)g Fj(m)32 b Fr(determinan)m(t,)i(w)m(e)f (plug)f(linear)f(form)m(ulas)h(of)h(size)g(at)h(most)f Fj(m)2676 1526 y Fm(2)2748 1559 y Fr(at)h(the)f(input)e(gates)j(of)g Fj( )s Fr(.)48 b(The)0 1672 y(resulting)28 b(size)h(is)f(at)j(most)e Fj(m)1047 1639 y Fm(2)1087 1672 y Fj(s)1130 1687 y Fm(det)1232 1672 y Fr(\()p Fj(m)p Fr(\).)40 b(Moreo)m(v)m(er)32 b(the)e(en)m(tries) f(of)h Fj(A)2463 1639 y Ff(0)2463 1698 y Fe(\016)2530 1672 y Fr(a\013ect)h(only)e(the)g(co)s(e\016cien)m(ts,)i(not)f(the)0 1785 y(structure,)g(of)h(the)f(resulting)f(form)m(ula.)p 3820 1729 80 7 v 3820 1795 7 67 v 3893 1795 V 3820 1802 80 7 v 141 2095 a(When)i Fj(m)25 b Fr(=)h Fj(n)664 2062 y Fe(O)r Fm(\(1\))844 2095 y Fr(or)31 b(ev)m(en)g Fj(m)26 b Fr(=)g Fj(n)1421 2062 y Fm(\(log)12 b Fe(n)p Fm(\))1620 2039 y Fc(O)r Fh(\(1\))1756 2095 y Fr(,)31 b Fj(m)1892 2062 y Fm(2)1931 2095 y Fj(s)1974 2110 y Fm(det)2076 2095 y Fr(\()p Fj(m)p Fr(\))26 b(=)g Fj(n)2404 2062 y Fm(\(log)13 b Fe(n)p Fm(\))2604 2039 y Fc(O)r Fh(\(1\))2765 2095 y Fr(=)30 b(size)h(quasip)s(olynomial)c(in)i Fj(n)p Fr(.)0 2208 y(Th)m(us)h(there)g(is)g(nearly)g(an)g(equiv)-5 b(alence)30 b(here)h(b)s(et)m(w)m(een)g Fj(f)2041 2175 y Fe(\036)2086 2208 y Fr(-stabilit)m(y)f(of)g(det)2701 2222 y Fe(m)2798 2208 y Fr(and)g(co)s(e\016cien)m(t-wise)h(appro)m(x-)0 2321 y(imabilit)m(y)d(of)i Fj(f)40 b Fr(b)m(y)30 b(form)m(ulae)g(of)h (quasi-p)s(olynomial)c(size,)j(for)g(an)m(y)h Fj(f)40 b Fr(not)30 b(just)g(the)h(\(padded\))f(p)s(ermanen)m(t.)141 2434 y(What)42 b(is)e(un)m(treated)h(in)f(\(b\))h(is)f(ho)m(w)h(the)g (magnitudes)f(of)h(constan)m(ts)h(in)d Fj(A)2886 2449 y Fe(\016)2965 2434 y Fr(dep)s(end)h(on)g Fj(\016)s Fr(,)45 b(hence)c(on)0 2547 y Fj(\017)p Fr(.)56 b(High)34 b(magnitudes)h(w)m (ould)f(prev)m(en)m(t)i(a)f(solid)f(link)f(b)s(eing)h(dra)m(wn)h(to)h Fi(c)-5 b(omputing)44 b Fr(appro)m(ximations)35 b(to)h(the)0 2660 y(p)s(ermanen)m(t)f(via)h(T)-8 b(uring)34 b(mac)m(hines)h(or)h (RAM)g(mo)s(dels)f(with)f(\\fair-cost)j(arithmetic.")57 b(Ho)m(w)m(ev)m(er,)39 b(algebraic)0 2773 y(complexit)m(y)34 b(theory)g(has)g(mainly)f(dev)m(elop)s(ed)g(to)i(ignore)f(the)g (magnitudes)f(of)h(constan)m(ts|see)i([Lok95)r(])e(as)h(a)0 2886 y(stem)d(pap)s(er)f(for)g(b)s(ounded-magnitude)e(lo)m(w)m(er)j(b)s (ounds)e(and)h(Mulm)m(uley's)f(o)m(wn)i(prior)e(w)m(ork)i([Mul99])g (for)g(the)0 2998 y(related)c(issue)e(of)i(constraining)f(access)i(to)f (individual)c(bits)i(of)i(constan)m(ts.)41 b(The)28 b(in)m(tuition)d (leading)i(V)-8 b(alian)m(t)28 b(to)0 3111 y(conjecture)j(exp)s(onen)m (tial)f(form)m(ula)f(size)h(lo)m(w)m(er)h(b)s(ounds)d(for)i(p)s(erm) 2328 3133 y Fe(n)2406 3111 y Fr(o)m(v)m(er)h(an)m(y)g(\014elds)e(not)i (of)f(c)m(haracteristic)h(2,)0 3224 y(not)c(caring)f(ab)s(out)g(the)h (constan)m(ts,)i(certainly)d(seems)h(to)g(extend)g(to)g(in)m(tuition)d (against)j(suc)m(h)f(appro)m(ximabilit)m(y)0 3337 y(b)m(y)k(quasip)s (olynomial-size)d(form)m(ulas.)40 b(Hence)31 b(this)f(bac)m(ks)g(up)g (Mulm)m(uley-Sohoni's)0 3548 y Fl(Conjecture)35 b(3.2)46 b Fi(The)34 b(p)-5 b(adde)g(d)37 b(p)-5 b(ermanent)36 b(do)-5 b(es)36 b(not)f(b)-5 b(elong)34 b(to)h(the)g(Z-closur)-5 b(e)34 b(of)h(the)f(SL)3321 3568 y Fe(m)3383 3549 y Fh(2)3422 3548 y Fi(-orbit)g(of)h(the)0 3661 y(determinant)f(in)f Fj(P)13 b Fr(\()p Fo(V)786 3675 y Fe(m)852 3661 y Fr(\))33 b Fi(under)g(any)h(p)-5 b(olynomial)35 b(\(or)f(sub-exp)-5 b(onential\))34 b(amount)g(of)f(p)-5 b(adding.)0 3872 y Fr(The)25 b($64,000)k(question)c(is)g(whether)g(this)g(reform)m (ulation)g(of)h(the)g(in)m(tuition)e(adds)h(something)g(tangibly)g(new) g(to)0 3985 y(the)31 b(basic)g(problem.)41 b(Historically)29 b(it)i(m)m(ust)g(b)s(e)f(remark)m(ed)h(that)h(in)e(the)h(1890s,)i (early)e(prop)s(onen)m(ts)f(of)h(group-)0 4098 y(represen)m(tation)d (theory)g(faced)h(detractors)g(suc)m(h)f(as)g(Burnside,)f(un)m(til)f(a) i(long)g(dev)m(elopmen)m(t)g(curv)m(e)h(\014nally)d(did)0 4211 y(start)31 b(pro)s(ducing)d(results)h(that)i(seemed)g (unobtainable)d(b)m(y)i(other)h(means.)141 4324 y(The)24 b(meat)g(of)h(the)f(matter)h(is)e(ho)m(w)h(w)m(ell)f(the)h(algebraic)f (to)s(ols)h(capture)g(complexit)m(y)g(prop)s(erties)e(of)i(b)s(oth)g (the)0 4437 y(function)f Fj(f)34 b Fr(b)s(eing)23 b(lo)m(w)m(er-b)s (ounded)f(and)i(the)h(\\univ)m(ersal")e(function)g(b)s(eing)g (orbited|here,)i(the)f(determinan)m(t.)0 4550 y(The)i(determinan)m(t)h (is)f(attractiv)m(ely)i(c)m(haracterizable)f(as)h(the)f(unique)e (degree-)p Fj(m)i Fr(homogeneous)h(function)d(that)0 4663 y(is)g(\014xed)g(b)m(y)g(a)h(certain)g(natural)e(group)h(action,)j (as)d(w)m(e)i(note)f(b)s(elo)m(w.)38 b(\(As)26 b(remark)m(ed)g(in)e ([MS02)q(],)j(other)f(families)0 4775 y(of)39 b(functions)e(that)i(are) g(similarly)c(univ)m(ersal)i(for)h(small)f(form)m(ulas)h(can)h(b)s(e)f (used,)i(if)d(they)i(ha)m(v)m(e)h(ev)m(en)f(nicer)0 4888 y(stabilizer)34 b(c)m(haracterizations.\))59 b(So)36 b(is)f(the)h(p)s(ermanen)m(t.)57 b(The)35 b(question)h(is)f(ho)m(w)h(w) m(ell)e(w)m(e)j(can)f(get)h(a)f(direct)0 5001 y(understanding)28 b(of)i(the)h(group)f(actions)g(on|and)g(in)f(the)h(neigh)m(b)s(orho)s (o)s(ds)e(of|these)j(functions.)141 5114 y(One)e(thing)g(that)h(can)g (help)e(in)g(a)i(direct)f(determination)f(of)i(whether)f(the)h(pro)5 b(jectiv)m(e)30 b(closure)f(of)g(an)h(orbit)0 5227 y(of)h(det)230 5241 y Fe(m)327 5227 y Fr(touc)m(hes)g(a)g(function)e Fj(f)40 b Fr(is)29 b(whether)h(the)h(a\016ne)f(orbit)g(of)g Fj(f)40 b Fr(itself)29 b(is)h(stable)g(\(under)f(the)i(same)f(group)0 5340 y(action\)!)39 b(If)22 b Fj(f)32 b Fr(is)22 b(stable,)j(then)d (kno)m(wn)h(algebraic)f(tec)m(hniques)h(come)g(in)m(to)g(pla)m(y)g(to)g (analyze)g(the)g(neigh)m(b)s(orho)s(o)s(ds)p eop %%Page: 9 9 9 8 bop 0 91 a Fr(of)24 b(the)h(orbits,)g(and)e(whether)h(the)g(orbits) f(approac)m(h)i(arbitrarily)d(closely)h(in)g(the)i (\(here-classical-is-equiv)-5 b(alen)m(t-)0 204 y(to-\))43 b(Zariski)c(sense.)73 b(Mulm)m(uley)39 b(and)i(Sohoni)f(observ)m(e)i (that)f(p)s(erm)2482 226 y Fe(m)2590 204 y Fr(is)f(stable,)k(but)d (unfortunately)-8 b(,)43 b(the)0 317 y(padded)34 b(p)s(ermanen)m(t)i(p) s(erm)988 284 y Fe(\036)988 340 y(n)1071 317 y Fr(is)f(unstable|not)f (ev)m(en)j(semi-stable.)56 b(Ho)m(w)m(ev)m(er|and)37 b(this)d(is)h(the)h(jumping-)0 430 y(p)s(oin)m(t)j(for)g(the)h(main)e (tec)m(hnical)h(con)m(ten)m(t)j(of)e(their)e(w)m(ork|they)i(dev)m(elop) f(a)h(notion)f(of)h(\\partial)f(stabilit)m(y")0 543 y(that)32 b(still)e(allo)m(ws)h(m)m(uc)m(h)g(of)h(the)g(desired)e(analytical)h (to)s(ols)g(to)i(b)s(e)e(reco)m(v)m(ered.)46 b(In)31 b(other)g(w)m(ords,)h(the)g(padding)0 656 y Fj(\036)g Fr(in)m(tro)s(duces)g(a)g(little)g(bit)f(of)i(\\nastiness")f(that)h (can)g(b)s(e)e(carefully)g(p)s(eeled)h(a)m(w)m(a)m(y)i(b)m(y)e (re\014ning)f(the)h(algebraic)0 769 y(analysis.)39 b(F)-8 b(or)31 b(this)f(w)m(e)h(need)f(more)g(de\014nitions.)0 1055 y Fk(4)135 b(P)l(artial)46 b(stabilit)l(y)0 1258 y Fr(Recall)36 b(that)h(a)g(parab)s(olic)d(subgroup)h Fj(P)48 b Fo(\022)35 b Fj(G)h Fr(stabilizes)f(a)i(\015ag)f(of)h(the)f (form)g(0)g Fo(\032)f Fj(W)3109 1272 y Fm(1)3183 1258 y Fo(\032)g Fj(:)15 b(:)g(:)37 b Fo(\032)e Fj(W)3623 1272 y Fe(r)3696 1258 y Fr(=)g Fj(V)20 b Fr(.)0 1371 y Fj(P)46 b Fr(is)33 b Fi(maximal)44 b Fr(if)33 b(it)g(is)f(not)i(con)m (tained)f(in)f(an)m(y)i(other)g(parab)s(olic)d(subgroup.)48 b(Again)34 b(follo)m(wing)e([AB95)q(])i(\(for)0 1484 y(represen)m(tation-tailored)25 b(rather)g(than)g(\\innate")h (de\014nitions\),)e(the)i Fi(unip)-5 b(otent)29 b(r)-5 b(adic)g(al)37 b Fr(of)26 b Fj(P)38 b Fr(is)24 b(the)i(subgroup)0 1597 y Fj(U)62 1611 y Fe(P)152 1597 y Fr(of)31 b Fj(P)44 b Fr(that)31 b(p)s(oin)m(t)m(wise)f(\014xes)g(ev)m(ery)i(quotien)m(t)f (subspace)g Fj(W)2230 1611 y Fe(i)2258 1597 y Fj(=W)2389 1611 y Fe(i)p Ff(\000)p Fm(1)2538 1597 y Fr(in)f(the)h(\015ag.)43 b Fj(U)3073 1611 y Fe(P)3162 1597 y Fr(is)30 b(alw)m(a)m(ys)i(a)f (normal)0 1710 y(subgroup.)38 b(A)26 b Fi(r)-5 b(e)g(ductive)34 b Fr(group)26 b(is)g(one)g(whose)g(represen)m(tation)h(forms)f(a)h (connected)g(set)g(of)g(p)s(oin)m(ts)e(\(unrolling)0 1823 y(the)k(represen)m(ting)e(matrices)i(as)f(v)m(ectors\))j(and)c (whose)i(unip)s(oten)m(t)e(radical)g(is)g(the)i(iden)m(tit)m(y)-8 b(.)40 b(F)-8 b(or)29 b(subgroups)e(of)0 1936 y Fj(P)13 b Fr(,)28 b(in)m(tuitiv)m(ely)e(this)h(means)h(that)g(the)g(reductiv)m (e)g(subgroup)e(p)s(oses)h(no)h(blo)s(c)m(k)-5 b(age)28 b(for)g(w)m(orking)f(do)m(wn)g(the)h(\015ag.)0 2049 y(The)g(iden)m(tit) m(y)g(group)g(itself)f(is)h(not)g(coun)m(ted)h(as)g(reductiv)m(e.)40 b(If)28 b(w)m(e)h(write)f Fj(W)2658 2063 y Fe(i)2711 2049 y Fr(=)d Fj(W)2893 2063 y Fe(i)p Ff(\000)p Fm(1)3028 2049 y Fr(+)16 b Fj(Y)3168 2063 y Fe(i)3224 2049 y Fr(for)28 b(eac)m(h)i Fj(i)p Fr(,)f(then)g(a)0 2161 y(maxim)m(um)f(subgroup)f Fj(L)878 2175 y Fe(P)965 2161 y Fr(of)i Fj(P)41 b Fr(that)30 b(stabilizes)d(all)h(the)g Fj(Y)2082 2175 y Fe(i)2139 2161 y Fr(is)g(called)g(a)h Fi(L)-5 b(evi)31 b(sub)-5 b(gr)g(oup)35 b Fr(or)29 b Fi(L)-5 b(evi)31 b(c)-5 b(omplement)0 2274 y Fr(of)26 b Fj(U)161 2288 y Fe(P)219 2274 y Fr(.)39 b(Suc)m(h)25 b(an)h Fj(L)682 2288 y Fe(P)766 2274 y Fr(is)e(alw)m(a)m (ys)i(isomorphic)e(to)i(the)g(direct)f(pro)s(duct)f Fi(GL)2568 2288 y Fe(y)2603 2297 y Fh(1)2642 2274 y Fr(\()p Fj(F)13 b Fr(\))d Fo(\002)g Fj(:)15 b(:)g(:)e Fo(\002)d Fi(GL)3201 2288 y Fe(y)3236 2296 y Fc(r)3274 2274 y Fr(\()p Fj(F)k Fr(\),)27 b(where)e(eac)m(h)0 2387 y Fj(y)45 2401 y Fe(i)98 2387 y Fr(=)g(dim)n(\()p Fj(Y)433 2401 y Fe(i)462 2387 y Fr(\))g(=)g(dim)o(\()p Fj(W)891 2401 y Fe(i)919 2387 y Fr(\))17 b Fo(\000)g Fr(dim)n(\()p Fj(W)1331 2401 y Fe(i)p Ff(\000)p Fm(1)1450 2387 y Fr(\).)40 b(The)29 b(p)s(oin)m(t)e(is)h(that)h Fj(P)39 b Fr(=)25 b Fj(U)2511 2401 y Fe(P)2569 2387 y Fj(L)2631 2401 y Fe(P)2718 2387 y Fr(and)k Fj(U)2956 2401 y Fe(P)3031 2387 y Fo(\\)17 b Fj(L)3171 2401 y Fe(P)3258 2387 y Fr(is)27 b(the)i(iden)m(tit)m(y)-8 b(,)30 b(a)0 2500 y(situation)g(summarized)g(b)m(y)h(sa)m(ying)g(that)h Fj(P)44 b Fr(is)30 b(the)h Fi(semi-dir)-5 b(e)g(ct)34 b(pr)-5 b(o)g(duct)42 b Fr(of)32 b(the)f(normal)f(subgroup)g Fj(U)3664 2514 y Fe(P)3754 2500 y Fr(and)0 2613 y(the)h(subgroup)d Fj(L)613 2627 y Fe(P)672 2613 y Fr(.)141 2726 y(The)44 b Fi(r)-5 b(ank)56 b Fr(of)45 b(an)g(algebraic)f(group)h Fj(G)f Fr(is)g(the)h(maxim)m(um)f(dimension)e(of)j(a)g(subgroup)e(of)i Fj(G)f Fr(that)i(is)0 2839 y(isomorphic)32 b(to)i(a)g(group)f(of)h (diagonal)f(matrices.)50 b(F)-8 b(or)34 b(example,)h(the)e(rank)h(of)f Fi(GL)2960 2853 y Fe(n)3007 2839 y Fr(\()p Fj(F)13 b Fr(\))34 b(is)f Fj(n)p Fr(,)h(but)f(the)g(rank)0 2952 y(of)k Fi(SL)218 2966 y Fe(n)265 2952 y Fr(\()p Fj(F)14 b Fr(\))37 b(is)f(only)g Fj(n)24 b Fo(\000)h Fr(1)37 b(since)f(one)i(of)f(the)g(diagonal)f(elemen)m(ts)i(is)e(constrained)g (b)m(y)h(the)g(pro)s(duct)f(of)h(the)0 3065 y(others)31 b(to)h(mak)m(e)g(the)g(determinan)m(t)e(equal)h(to)h(1.)43 b(Finally)-8 b(,)30 b(a)i(subgroup)d(of)i(an)g(algebraic)g(group)g Fj(G)f Fr(is)h Fi(r)-5 b(e)g(gular)0 3178 y Fr(if)32 b(its)h Fi(r)-5 b(o)g(ot)38 b(system)j Fr(is)33 b(a)g(subsystem)g(of)g (that)i(pf)d Fj(G)p Fr(.)50 b(\(W)-8 b(e)35 b(do)e(not)h(go)g(in)m(to)f (ro)s(ot)h(systems)g(here,)g(but)f(see)h(the)0 3291 y(App)s(endix)f(of) k([Hum81].\))58 b(Although)36 b(in)m(tuition)d(in)i(a)i(short)e (treatmen)m(t)j(has)e(to)h(lag)f(b)s(ehind)d(here,)k(w)m(e)g(no)m(w)0 3403 y(ha)m(v)m(e)32 b(all)d(the)h(de\014nitions)e(required)h(to)i (de\014ne)f(partial)f(stabilit)m(y)-8 b(.)0 3616 y Fl(De\014nition)35 b(4.1)h(\([MS02]\).)45 b Fr(A)37 b(pro)5 b(jectiv)m(e)38 b(p)s(oin)m(t)e Fj(y)j Fo(2)c Fj(P)13 b Fr(\()p Fj(V)21 b Fr(\))37 b(is)f Fi(p)-5 b(artial)5 b(ly)41 b(stable)e(with)h(defe)-5 b(ct)38 b Fj(\016)n(;)15 b Fr(\001)38 b(with)0 3729 y(resp)s(ect)30 b(to)h(the)f(action)h(of)f(an)g(algebraic)g(group)g Fj(G)f Fr(if)g(there)i(are)f(a)h(maximal)e(parab)s(olic)f(subgroup)h Fj(P)13 b Fr(,)30 b(a)h(Levi)0 3842 y(subgroup)d Fj(L)g Fr(of)i Fj(P)13 b Fr(,)29 b(and)g(a)h(regular)e(reductiv)m(e)h (subgroup)f Fj(K)35 b Fr(of)30 b Fj(L)f Fr(suc)m(h)g(that)g(for)g(all)f (a\016ne)i(p)s(oin)m(ts)e Fj(x)d Fo(2)g Fj(V)49 b Fr(on)0 3955 y(the)31 b(line)d Fj(y)s Fr(:)66 4142 y(\(a\))46 b(The)30 b(isotrop)m(y)g(subgroup)f Fj(G)1230 4156 y Fe(x)1304 4142 y Fr(con)m(tains)h(the)h(unip)s(oten)m(t)e(radical)g Fj(U)2591 4156 y Fe(P)2650 4142 y Fr(;)61 4330 y(\(b\))45 b Fj(L)20 b Fo(\\)g Fj(G)462 4344 y Fe(x)536 4330 y Fr(is)29 b(reductiv)m(e;)71 4518 y(\(c\))46 b Fj(y)33 b Fr(is)d(stable)g(under)f (the)h(action)h(b)m(y)f Fj(K)7 b Fr(;)61 4705 y(\(d\))45 b Fi(r)-5 b(ank)11 b Fr(\()p Fj(K)c Fr(\))26 b(=)f Fi(r)-5 b(ank)11 b Fr(\()p Fj(L)p Fr(\))20 b Fo(\000)g Fj(\016)s Fr(;)32 b(and)71 4893 y(\(e\))46 b(dim)o(\()p Fj(Lx)p Fr(\))p Fj(=)15 b Fr(dim)o(\()p Fj(Gx)p Fr(\))29 b Fo(\025)f Fr(\001,)k(i.e.,)h(the)g(dimension)d(of)i(the)g(orbit)g Fj(Lx)g Fr(is)f(su\016cien)m(tly)g(large)h(compared)g(to)227 5006 y(the)f(dimension)d(of)i(the)h(orbit)e Fj(Gx)p Fr(.)0 5218 y(If)38 b Fj(P)53 b Fr(=)39 b Fj(G)p Fr(,)i Fj(\016)i Fr(=)c(0,)j(and)c(\001)h(=)g(1,)j(then)c(one)h(gets)h Fj(K)46 b Fr(=)40 b Fj(L)f Fr(=)g Fj(G)f Fr(also,)j(and)e(b)m(y)f (\(c\))i(this)e(co-incides)g(with)0 5331 y(the)c(de\014nition)e(of)i Fj(y)j Fr(b)s(eing)32 b(stable)i(under)e Fj(G)p Fr(.)51 b(Th)m(us)33 b Fj(\016)38 b Fr(and)33 b(\001)h(quan)m(tify)f(the)h (deviation)g(from)f(stabilit)m(y)-8 b(.)51 b(In)p eop %%Page: 10 10 10 9 bop 0 91 a Fr(the)37 b(cases)g(of)g(partial)e(stabilit)m(y)g (sough)m(t)i(and)f(used)g(b)m(y)g(Mulm)m(uley)f(and)h(Sohoni,)h Fj(\016)i Fr(=)c(1)i(and)f(\001)g(is)g(in)m(v)m(erse-)0 204 y(p)s(olynomial)21 b(in)h Fj(n)p Fr(,)j(so)f(the)f(deviation)g(is)f (\\not)j(to)s(o)f(large.")39 b(The)23 b(in)m(ten)m(t,)i(to)g(b)s(e)e (brough)m(t)g(out)h(in)e(the)i(companion)0 317 y(pap)s(er)29 b([MS02b)q(],)i(is)e(to)i(fo)s(cus)f(in)f(on)h(the)h(action)g(of)f Fj(K)7 b Fr(.)0 518 y Fl(Theorem)34 b(4.1)h(\(Theorem)f(4.3)h(in)g ([MS02)q(]\))45 b Fi(The)31 b(p)-5 b(adde)g(d)33 b(p)-5 b(ermanent)33 b Fr(p)s(erm)2925 485 y Fe(\036)2925 540 y(n)3002 518 y Fi(as)e(a)g(memb)-5 b(er)31 b(of)g Fj(P)13 b Fr(\()p Fo(V)3798 532 y Fe(m)3865 518 y Fr(\))0 630 y Fi(with)37 b Fj(m)281 597 y Fm(2)357 630 y Fi(variables)g(is)g(p)-5 b(artial)5 b(ly)38 b(stable)f(under)g(the)g(action)g(by)g(SL)2388 650 y Fe(m)2450 631 y Fh(2)2489 630 y Fr(\()p Fj(F)13 b Fr(\))p Fi(,)38 b(with)f(defe)-5 b(ct)37 b Fj(\016)f Fr(=)c(1)k Fi(and)i Fr(\001)e Fi(p)-5 b(oly-)0 743 y(nomial)35 b(in)e Fj(m=n)p Fi(.)44 b(This)34 b(applies)h(to)f(any)g(p)-5 b(adde)g(d)36 b(homo)-5 b(gene)g(ous)36 b(form)f Fj(h)e Fi(of)h(de)-5 b(gr)g(e)g(e)34 b Fj(n)27 b(<)f(m)33 b Fi(in)h Fj(n)3455 710 y Fm(2)3527 743 y Fi(variables,)0 856 y(such)f(that)h Fj(h)f Fi(is)f(stable)h(as)g(a)g(memb)-5 b(er)34 b(of)f Fj(P)13 b Fr(\()p Fo(V)1633 870 y Fe(n)1680 856 y Fr(\))33 b Fi(under)g(the)g(action)g(by)g(SL)2661 876 y Fe(n)2704 857 y Fh(2)2742 856 y Fi(.)0 1141 y Fk(5)135 b(Obstructions)0 1344 y Fr(Giv)m(en)40 b(p)s(olynomials)e Fj(f)49 b Fr(and)40 b Fj(g)k Fr(suc)m(h)c(as)h(p)s(erm)1699 1311 y Fe(\036)1785 1344 y Fr(and)f(det)2098 1358 y Fe(m)2165 1344 y Fr(,)j(resp)s(ectiv)m(ely)-8 b(,)43 b(and)d(a)g(relev)-5 b(an)m(t)41 b(action)g(b)m(y)f(an)0 1457 y(algebraic)c(group)f Fj(G)p Fr(,)i(an)e Fi(obstruction)44 b Fr(is)35 b(a)h(witness)f(that)h Fj(f)45 b Fr(do)s(es)36 b(not)g(lie)e(in)h(the)h(closure)f(of)h(the)g (orbit)f Fj(Gg)0 1570 y Fr(in)j Fj(P)13 b Fr(\()p Fo(V)7 b Fr(\).)68 b(This)37 b(witness)i(can)g(b)s(e)g(a)g(\\meta-p)s (olynomial")g Fj(q)s Fr(.)67 b(Namely)-8 b(,)42 b(if)c Fj(f)49 b Fr(and)38 b Fj(g)43 b Fr(are)d(homogeneous)g(of)0 1682 y(degree)35 b Fj(d)f Fr(in)e Fj(r)37 b Fr(v)-5 b(ariables)32 b(\(ab)s(o)m(v)m(e,)37 b Fj(d)31 b Fr(=)g Fj(m)j Fr(and)f Fj(r)h Fr(=)d Fj(m)1989 1649 y Fm(2)2028 1682 y Fr(\),)k(then)f Fj(f)43 b Fr(and)33 b Fj(g)38 b Fr(are)c(p)s(oin)m(ts)f(in)f(a)j(v)m (ector)g(space)g(of)0 1795 y Fj(R)c Fr(=)f(\()251 1757 y Fe(m)p Fm(+)p Fe(d)p Ff(\000)p Fm(1)252 1823 y Fe(d)515 1795 y Fr(\))j(dimensions|as)e(are)j(all)f(p)s(olynomials)d(in)i(the)i (orbit)e Fj(Gg)37 b Fr(and)c(its)g(pro)5 b(jectiv)m(e)34 b(Z-closure.)49 b(A)0 1908 y(p)s(olynomial)28 b Fj(q)34 b Fr(in)c Fj(R)h Fr(v)-5 b(ariables)30 b(can)h(th)m(us)f(b)s(e)h(said)f (to)h(tak)m(e)i(these)e(smaller)f(p)s(olynomials)e(as)j(argumen)m(ts.) 43 b(If)30 b Fj(q)0 2021 y Fr(v)-5 b(anishes)24 b(on)g(all)g(p)s(oin)m (ts)g(in)f Fj(Gg)s Fr(,)k(then)d(it)g(also)h(v)-5 b(anishes)24 b(on)h(the)g(closure.)38 b(F)-8 b(urther,)26 b(if)e(suc)m(h)g(a)h Fj(q)j Fr(giv)m(es)d Fj(q)s Fr(\()p Fj(f)10 b Fr(\))25 b Fo(6)p Fr(=)g(0,)0 2134 y(then)33 b Fj(q)i Fr(witnesses)d(that)i Fj(f)42 b Fr(do)s(es)33 b(not)g(b)s(elong)f(to)i(the)f(orbit)f (closure.)48 b(Ho)m(w)m(ev)m(er,)36 b(suc)m(h)c(a)h(witness)f(could)g (b)s(e)h(a)0 2247 y(double-exp)s(onen)m(tial-size)e(ob)5 b(ject,)34 b(and)e(it)g(is)f(not)i(clear)f(that)h(w)m(e)g(ha)m(v)m(e)h (gained)e(an)m(y)h(information)e(ab)s(out)h(the)0 2360 y(\()p Fj(f)5 b(;)15 b(g)s Fr(\)-problem)30 b(b)m(y)g(doing)g(this.)141 2473 y(Rather,)h(Mulm)m(uley)e(and)g(Sohoni)g(adv)-5 b(ance)31 b(the)g(goal)f(of)h(constructing)f(represen)m(tations)g(of)h (the)f(isotrop)m(y)0 2586 y(subgroups)d Fj(G)501 2601 y Fe(f)574 2586 y Fr(and)h Fj(G)821 2600 y Fe(g)889 2586 y Fr(that)h(serv)m(e)g(as)g(a)g(witness.)38 b(\(In)29 b([MS02)q(],)g Fj(G)2396 2601 y Fe(f)2469 2586 y Fr(is)f(called)f(\\)p Fj(H)7 b Fr(")30 b(and)d Fj(G)3263 2600 y Fe(g)3331 2586 y Fr(is)h(called)g(\\)p Fj(Q)p Fr(."\))0 2786 y Fl(Theorem)34 b(5.1)h(\(Theorem)f(5.1)h(in)g([MS02)q(]\))45 b Fi(L)-5 b(et)38 b Fj(f)46 b Fi(b)-5 b(e)36 b(stable)i(under)f(the)g(action)h (of)f(a)g(gr)-5 b(oup)38 b Fj(G)f Fi(on)g Fo(V)7 b Fi(.)0 2899 y(A)35 b(nonzer)-5 b(o)36 b(r)-5 b(epr)g(esentation)38 b Fo(W)k Fi(of)36 b Fj(G)e Fi(is)h(an)h(obstruction)g(for)g Fr(\()p Fj(f)5 b(;)15 b(g)s Fr(\))36 b Fi(if)e(its)i(unique)e (factorization)j(into)f(irr)-5 b(e-)0 3012 y(ducibles)39 b(c)-5 b(ontains)40 b(an)g(o)-5 b(c)g(curr)g(enc)g(e)40 b(of)f(the)h(trivial)g Fj(G)1921 3027 y Fe(f)1965 3012 y Fi(-mo)-5 b(dule)40 b Fj(\034)2353 3027 y Fe(f)2437 3012 y Fi(but)f(not)h(a)f(trivial)h Fj(G)3196 3026 y Fe(g)3235 3012 y Fi(-mo)-5 b(dule)40 b Fj(\034)3623 3026 y Fe(g)3663 3012 y Fi(|and)0 3125 y(mor)-5 b(e)34 b(gener)-5 b(al)5 b(ly,)33 b(if)f Fj(\034)769 3140 y Fe(f)847 3125 y Fi(o)-5 b(c)g(curs)34 b(mor)-5 b(e)34 b(often)f(than)g Fj(\034)1830 3139 y Fe(g)1903 3125 y Fi(in)f(this)h(factorization.)0 3325 y Fr(This)e(reads)i(lik)m(e)f(a)h(de\014nition)e(but)h(is)g (actually)g(a)h(theorem,)h(replacing)e(\\is)g(an)h(obstruction")g(with) e(the)i(con-)0 3438 y(clusion,)f(\\then)h Fj(f)42 b Fr(is)32 b(not)h(in)f(the)h(pro)5 b(jectiv)m(e)33 b(closure)f(of)h Fj(Gg)s Fr(.")49 b(W)-8 b(e)34 b(can)f(w)m(ord)g(the)g(more-general)g (condition)0 3551 y(a)e(little)e(more)h(helpfully)-8 b(.)38 b(De\014ne)1148 3856 y Fj(E)1215 3871 y Fe(f)1343 3856 y Fr(=)83 b Fo(f)15 b Fj(v)29 b Fo(2)c(W)33 b Fr(:)25 b(\()p Fo(8)p Fj(A)g Fo(2)g Fj(G)2226 3871 y Fe(f)2271 3856 y Fr(\))h Fj(A)5 b Fo(\001)g Fj(v)29 b Fr(=)c Fj(v)18 b Fo(g)p Fj(;)1153 3994 y(E)1220 4008 y Fe(g)1343 3994 y Fr(=)83 b Fo(f)15 b Fj(v)29 b Fo(2)c(W)33 b Fr(:)25 b(\()p Fo(8)p Fj(A)g Fo(2)g Fj(G)2226 4008 y Fe(g)2266 3994 y Fr(\))g Fj(A)5 b Fo(\001)g Fj(v)30 b Fr(=)25 b Fj(v)18 b Fo(g)p Fj(:)0 4324 y Fr(Then)41 b Fj(E)316 4339 y Fe(f)403 4324 y Fr(and)h Fj(E)659 4338 y Fe(g)741 4324 y Fr(are)g(closed)g(under)f(scalar)h(m)m(ultiplication)d(and)i (under)f(addition,)k(so)e(they)g(are)h(linear)0 4437 y(subspaces|resp)s(ectiv)m(ely)-8 b(,)31 b(the)h(subspace)f(p)s(oin)m (t)m(wise)g(\014xed)g(b)m(y)g Fj(G)2354 4452 y Fe(f)2431 4437 y Fr(and)g(the)h(space)g(p)s(oin)m(t)m(wise)f(\014xed)g(b)m(y)g Fj(G)3835 4451 y Fe(g)3875 4437 y Fr(.)0 4550 y(The)f(more-general)g (condition)f(is)g(then)h(simply)e(dim)n(\()p Fj(E)1968 4565 y Fe(f)2014 4550 y Fr(\))e Fj(>)f Fr(dim)n(\()p Fj(E)2424 4564 y Fe(g)2465 4550 y Fr(\)|i.e.,)31 b(construction)e(of)i (a)f(represen)m(ta-)0 4663 y(tion)g Fj(W)43 b Fr(giving)29 b(this)g(implies)f(that)j Fj(f)39 b Fr(is)30 b(not)g(in)f(the)i(pro)5 b(jectiv)m(e)31 b(closure)f(of)g Fj(Gg)s Fr(.)141 4775 y(T)-8 b(o)25 b(pro)m(v)m(e)f(this,)h(note)f(\014rst)f(that)i(this)e (disparit)m(y)f(of)i(dimension)d(is)i(imp)s(ossible)d(if)j Fj(G)3006 4790 y Fe(f)3075 4775 y Fr(con)m(tains)h Fj(AG)3565 4789 y Fe(g)3605 4775 y Fj(A)3673 4742 y Ff(\000)p Fm(1)3791 4775 y Fr(for)0 4888 y(some)32 b Fj(A)27 b Fo(2)g Fj(G)p Fr(.)44 b(This)30 b(is)g(in)m(tuitiv)m(ely)g(b)s(ecause)h(dim)o(\()p Fj(E)1881 4903 y Fe(f)1927 4888 y Fr(\))c Fj(>)g Fr(dim)o(\()p Fj(E)2341 4902 y Fe(g)2381 4888 y Fr(\))32 b(sa)m(ys)g(that)g Fj(G)2912 4903 y Fe(f)2989 4888 y Fr(should)d(b)s(e)i(smaller)f(than)0 5001 y Fj(G)72 5015 y Fe(g)112 5001 y Fr(,)h(since)g(larger)g(groups)f (\014x)h(few)m(er)h(elemen)m(ts.)43 b(Notice)32 b(that)g Fj(AG)2347 5015 y Fe(g)2387 5001 y Fj(A)2455 4968 y Ff(\000)p Fm(1)2581 5001 y Fr(forms)e(a)i(group,)f(called)g(a)g Fi(c)-5 b(onjugate)0 5114 y Fr(of)31 b Fj(G)176 5128 y Fe(g)215 5114 y Fr(.)141 5227 y(Ho)m(w)m(ev)m(er,)g(the)d(stabilit)m (y)e(of)i Fj(f)37 b Fr(implies)24 b(b)m(y)k(a)g(result)f(kno)m(wn)g(as) h Fi(Luna's)i(slic)-5 b(e)31 b(the)-5 b(or)g(em)36 b Fr(that)29 b(the)e(orbit)g Fj(Gf)0 5340 y Fr(in)f Fo(V)34 b Fr(has)27 b(a)h(neigh)m(b)s(orho)s(o)s(d)c Fj(U)37 b Fr(preserv)m(ed)27 b(b)m(y)g Fj(G)p Fr(,)h(suc)m(h)e(that)i(the)f (isotrop)m(y)g(subgroup)f Fj(S)3083 5354 y Fe(p)3149 5340 y Fr(of)i(an)m(y)f(p)s(oin)m(t)f Fj(p)h Fr(in)f Fj(U)p eop %%Page: 11 11 11 10 bop 0 91 a Fr(is)28 b(a)h(conjugate)i(of)e(a)g(subgroup)e(of)i Fj(G)1318 106 y Fe(f)1363 91 y Fr(.)40 b(Let)30 b([)p Fo(\001)p Fr(])f(denote)h(the)f(mapping)e(from)i Fo(V)24 b(n)18 b(f)d Fr(0)g Fo(g)31 b Fr(to)e Fj(P)13 b Fr(\()p Fo(V)7 b Fr(\).)41 b(If)29 b([)p Fj(f)10 b Fr(])29 b(lies)e(in)0 204 y(the)f(\(Z-equiv)-5 b(alen)m(t-to-classical\))26 b(closure)f(of)h Fj(G)p Fr([)p Fj(g)s Fr(],)h(then)f([)p Fj(U)10 b Fr(])26 b(con)m(tains)g(a)g(p)s(oin)m(t)e([)p Fj(A)l Fo(\001)l Fj(g)s Fr(])j(for)e(some)h Fj(A)g Fo(2)e Fj(G)p Fr(.)39 b(Th)m(us)0 317 y Fj(U)45 b Fr(con)m(tains)35 b(a)g(p)s(oin)m(t)f Fj(p)f Fr(=)g Fj(\025A)8 b Fo(\001)g Fj(g)39 b Fr(with)34 b Fj(\025)f Fo(2)f Fj(F)1679 281 y Ff(\003)1754 317 y Fr(\(i.e.,)37 b Fj(\025)c Fo(6)p Fr(=)f(0\).)56 b(No)m(w)35 b(the)h(isotrop)m(y)e(subgroup)g Fj(S)3499 331 y Fe(p)3571 317 y Fr(=)f Fj(G)3747 332 y Fe(\025A)p Ff(\001)p Fe(g)0 430 y Fr(equals)28 b Fj(G)346 444 y Fe(A)p Ff(\001)p Fe(g)487 430 y Fr(for)h(an)m(y)h(nonzero)f (scalar)g Fj(\025)p Fr(.)40 b(No)m(w)30 b(observ)m(e)f(that)h Fj(G)2304 444 y Fe(A)p Ff(\001)p Fe(g)2441 430 y Fr(=)25 b Fj(AG)2677 444 y Fe(g)2717 430 y Fj(A)2785 397 y Ff(\000)p Fm(1)2879 430 y Fr(,)30 b(b)s(ecause)f(if)f Fj(B)7 b Fo(\001)r Fj(g)29 b Fr(=)c Fj(g)s Fr(,)30 b(then)730 629 y(\()p Fj(AB)5 b(A)975 591 y Ff(\000)p Fm(1)1069 629 y Fr(\))g Fo(\001)g Fr(\()p Fj(A)g Fo(\001)g Fj(g)s Fr(\))28 b(=)d(\()p Fj(AB)5 b(A)1727 591 y Ff(\000)p Fm(1)1822 629 y Fj(A)p Fr(\))g Fo(\001)g Fj(g)29 b Fr(=)c(\()p Fj(AB)5 b Fr(\))g Fo(\001)g Fj(g)30 b Fr(=)25 b Fj(A)5 b Fo(\001)g Fr(\()p Fj(B)10 b Fo(\001)5 b Fj(g)s Fr(\))27 b(=)e Fj(A)5 b Fo(\001)g Fj(g)s(:)0 828 y Fr(Th)m(us)36 b(elemen)m(ts)i(of)f Fj(AG)866 842 y Fe(g)906 828 y Fj(A)974 795 y Ff(\000)p Fm(1)1106 828 y Fr(are)g(precisely)f(those)i(that)g (\014x)f Fj(A)10 b Fo(\001)g Fj(g)s Fr(,)39 b(so)f Fj(S)2626 842 y Fe(p)2702 828 y Fr(is)f(a)g(conjugate)i(of)e Fj(G)3484 842 y Fe(g)3524 828 y Fr(.)61 b(Since)37 b(a)0 940 y(conjugate)f(of)e (a)h(conjugate)g(is)f(a)g(conjugate,)j(Luna's)d(slice)f(theorem)i (tells)e(us)h(that)h Fj(G)3083 954 y Fe(g)3156 940 y Fr(is)f(a)h(conjugate)g(of)g(a)0 1053 y(subgroup)c(of)i Fj(G)575 1068 y Fe(f)620 1053 y Fr(.)48 b(This)31 b(is)h(exactly)h (what)g(w)m(e)g(argued)g(couldn't)f(happ)s(en)f(from)h(the)h (dimensions)d(of)j Fj(E)3675 1068 y Fe(f)3754 1053 y Fr(and)0 1166 y Fj(E)67 1180 y Fe(g)138 1166 y Fr(in)c(the)h(represen)m (tation.)41 b(Hence)31 b([)p Fj(f)10 b Fr(])30 b(cannot)h(b)s(elong)f (to)h(the)f(closure)g(of)h(the)f(orbit)g Fj(G)p Fr([)p Fj(g)s Fr(])h(in)e Fj(P)13 b Fr(\()p Fo(V)7 b Fr(\).)141 1279 y(This)29 b(pro)s(of|and)h(its)g(exploitation)h(of)g(the)g (stabilit)m(y)f(of)h Fj(f)40 b Fr(itself|mo)m(v)m(es)31 b(the)h(fo)s(cus)e(on)m(to)i(constructing)0 1392 y(informativ)m(e,)40 b(extremal)e(represen)m(tations)g(of)h Fj(G)p Fr(.)64 b(This)36 b(is)h(a)i(m)m(uc)m(h-studied)e(area)i(of)f(mathematics,)j (where)0 1505 y(man)m(y)f(kinds)e(of)i(demands)f(ha)m(v)m(e)i(already)e (b)s(een)h(dealt)f(with.)68 b(The)40 b(demands)e(here)i(are)h(go)m(v)m (erned)g(largely)0 1618 y(b)m(y)c(the)h(structure)f(of)g(the)h(isotrop) m(y)f(subgroups)e Fj(G)1833 1633 y Fe(f)1915 1618 y Fr(and)i Fj(G)2171 1632 y Fe(g)2211 1618 y Fr(,)i(whic)m(h)d(in)g(the)h(case)i (of)e(the)h(p)s(ermanen)m(t)f(and)0 1731 y(the)h(determinan)m(t)f(ha)m (v)m(e)i(b)s(een)e(nicely)g(c)m(haracterized)i(ab)s(o)m(v)m(e.)64 b(Moreo)m(v)m(er,)42 b(a)c(ma)5 b(jor)38 b(metho)s(dological)f(p)s(oin) m(t)0 1844 y(adv)-5 b(anced)26 b(in)f([MS02)q(])i(is)e(that)i(the)f(fo) s(cus)g(can)h(shift)e(from)h Fj(f)35 b Fr(and)26 b Fj(g)k Fr(to)d(their)e(isotrop)m(y)h(subgroups)e(themselv)m(es.)0 1957 y(Those)k(computed)h(at)g(the)g(end)f(of)h(Section)g(2.1)g(here)g (for)f(p)s(erm)2219 1978 y Fe(n)2295 1957 y Fr(and)g(det)2596 1971 y Fe(n)2672 1957 y Fr(in)m(v)m(olv)m(e)h(b)s(edro)s(c)m(k)f (matrix)g(notions)0 2070 y(that)k(do)f(not)h(immediately)e(shout,)i (\\I)f(am)h(merely)f(a)h(re-co)s(ding)f(of)g(the)h(p)s(ermanen)m(t)f (or)g(determinan)m(t!")44 b(The)0 2182 y(question)27 b(of)g(whether)g Fj(f)36 b Fr(is)27 b(in)f(the)h(pro)5 b(jectiv)m(e)28 b(orbit)e(closure)h(of)h Fj(Gg)i Fr(can)e(b)s(e)e (reform)m(ulated)h(as)h(the)f(\\stabilizer)0 2295 y(problem")35 b(of)h(whether)f Fj(G)940 2309 y Fe(g)1015 2295 y Fr(can)i(o)s(ccur)e (as)h(an)g(isotrop)m(y)g(subgroup)e(in)h(arbitrarily-close)e(neigh)m(b) s(orho)s(o)s(ds)h(of)0 2408 y(the)d(orbit)e(of)i(items)f(\014xed)f(b)m (y)h Fj(G)1143 2423 y Fe(f)1188 2408 y Fr(.)141 2521 y(Ah,)36 b(but)f(Conjecture)g(3.2)h(references)f(the)g Fi(p)-5 b(adde)g(d)48 b Fr(p)s(ermanen)m(t.)54 b(P)m(artial)35 b(stabilit)m(y)e(do)s(es)i(not)g(yield)f(the)0 2634 y(e\016cien)m(t)j (conclusion)e(of)h(Luna's)g(slice)f(theorem.)59 b(Mulm)m(uley)35 b(and)g(Sohoni)g(giv)m(e)i(one)g(adaptation)f(of)h(Theo-)0 2747 y(rem)31 b(5.1)i Fi(towar)-5 b(d)43 b Fr(the)32 b(partially-stable)d(case,)k(as)f(\\Theorem)g(5.3")h(in)d(the)h(part-I) h(pap)s(er)e([MS02)q(],)i(but)f(it)g(still)0 2860 y(requires)24 b Fj(f)34 b Fr(to)25 b(b)s(e)g(stable.)38 b(The)25 b(full)d(dev)m (elopmen)m(t)k(of)f(obstructions)f(in)g(the)h(partially-stable)e(case)j (is)e(the)h(main)0 2973 y(sub)5 b(ject)36 b(of)g(the)h(part-I)s(I)e (pap)s(er)h([MS02b],)i(to)f(app)s(ear.)58 b(According)36 b(to)h(the)f(o)m(v)m(erview)h(pap)s(er)e([MS02c)r(],)j(this)0 3086 y(pro)s(ceeds)25 b(deep)s(er)g(in)m(to)h(cen)m(tury-old)g(unsolv)m (ed)e(problems)h(in)f(represen)m(tation)i(theory)g(suc)m(h)f(as)h(the)g (\\pleth)m(ysm)0 3199 y(problem.")39 b(A)m(t)30 b(this)d(p)s(oin)m(t,)h (as)h(complexit)m(y)g(theorists)f(w)m(e)h(should)d(\014rst)i(tak)m(e)j (one)d(step)h(bac)m(k)h(to)f(try)f(to)i(assess)0 3312 y(the)h(com)m(binatorial)e(nature)h(of)h(the)f(pro)s(of)g(tec)m (hniques)g(and)g(problems)e(encoun)m(tered.)0 3597 y Fk(6)135 b(W)-11 b(ould)44 b(this)i(naturalize?)0 3800 y Fr(Razb)s(oro)m(v)27 b(and)f(Rudic)m(h)f([RR97)q(])i(sho)m(w)m(ed)f (the)h(existence)g(of)f(a)h(new)f(obstacle)h(to)g(circuit)f(lo)m(w)m (er-b)s(ound)f(pro)s(ofs.)0 3913 y(They)e(observ)m(ed)h(that)g (basically)e(all)h(kno)m(wn)g(pro)s(ofs)g(that)h(certain)g(Bo)s(olean)g (functions)e Fj(h)3086 3927 y Fe(n)3157 3913 y Fr(lie)g(outside)h(a)h (circuit)0 4026 y(class)e Fo(C)28 b Fr(rev)m(olv)m(e)23 b(around)f(sequences)g(\005)1354 4040 y Fe(n)1424 4026 y Fr(of)g(subsets)g(of)h Fo(F)1987 4040 y Fe(n)2056 4026 y Fr(\(the)g(set)g(of)g(2)2515 3993 y Fm(2)2550 3969 y Fc(n)2620 4026 y Fr(Bo)s(olean)f(functions)f(of)i Fj(n)f Fr(v)-5 b(ariables\))0 4139 y(suc)m(h)30 b(that:)66 4322 y(\(a\))46 b(no)32 b(language)g Fj(L)f Fr(suc)m(h)g(that)h Fj(L)1294 4289 y Fm(=)p Fe(n)1423 4322 y Fo(2)27 b Fr(\005)1579 4336 y Fe(n)1658 4322 y Fr(for)k(almost)g(all)g(\(v)-5 b(arian)m(tly)d(,)32 b(in\014nitely-man)m(y\))d Fj(n)i Fr(b)s(elongs)g(to)h Fo(C)227 4435 y Fr(\(\\usefulness"\);)61 4621 y(\(b\))45 b Fo(j)p Fr(\005)320 4635 y Fe(n)368 4621 y Fo(j)p Fj(=)p Fo(jF)528 4635 y Fe(n)576 4621 y Fo(j)31 b Fr(is)e(b)s(ounded)f(b)s(elo)m(w)i(b)m(y)g(1)p Fj(=p)p Fr(\(2)1694 4588 y Fe(n)1742 4621 y Fr(\))h(for)f(some)h(p)s (olynomial)d Fj(p)i Fr(\(\\largeness"\);)71 4818 y(\(c\))46 b(whether)31 b(a)g(giv)m(en)g Fj(f)36 b Fo(2)26 b(F)1124 4832 y Fe(n)1202 4818 y Fr(b)s(elongs)k(to)i(\005)1709 4832 y Fe(n)1787 4818 y Fr(is)e(decidable)f(in)h(time)h(2)2638 4785 y Fe(n)2681 4762 y Fc(O)r Fh(\(1\))2816 4818 y Fr(,)g(whic)m(h)f (is)g(quasi-p)s(olynomial)227 4931 y(in)f(the)i(length)f(2)813 4898 y Fe(n)890 4931 y Fr(of)h(the)g(truth)e(table)h(of)h Fj(f)40 b Fr(giv)m(en)30 b(as)h(input)d(\(\\lo)m(w)j(complexit)m(y"\);) 0 5114 y(and)i Fj(h)232 5128 y Fe(n)311 5114 y Fo(2)e Fr(\005)471 5128 y Fe(n)552 5114 y Fr(for)j(all)f Fj(n)p Fr(.)51 b(Razb)s(oro)m(v)35 b(and)e(Rudic)m(h)g(sho)m(w)m(ed)h(that)h (if)e(a)h(sequence)h(\005)2997 5128 y Fe(n)3078 5114 y Fr(satis\014es)f(\(a\){\(c\))i(with)0 5227 y Fo(C)49 b Fr(=)43 b(P)p Fj(=)p Fi(p)-5 b(oly)51 b Fr(\(i.e.,)45 b(the)c(class)g(of)g(languages)h(ha)m(ving)f(p)s(olynomial-sized)d (circuits\),)43 b(then)e(pseudo-random)0 5340 y(generators)29 b(and)f(one-w)m(a)m(y)h(functions)e(of)h(exp)s(onen)m(tial)g(securit)m (y)g(do)g(not)g(exist.)40 b(Since)27 b(the)h(factoring)h(problem)p eop %%Page: 12 12 12 11 bop 0 91 a Fr(is)38 b(widely)f(b)s(eliev)m(ed)g(to)j(b)s(e)e (hard)g(enough)g(to)i(pro)s(duce)e(suc)m(h)g(generators,)k(suc)m(h)d (\\natural)f(pro)s(ofs")h(\005)3693 105 y Fe(n)3779 91 y Fr(are)0 204 y(conjectured)22 b(not)g(to)h(exist.)37 b(Gran)m(ting)22 b(this,)h(a)f(pro)s(of)f(that)i(NP-complete)f (problems)e(do)i(not)g(ha)m(v)m(e)h(p)s(olynomial-)0 317 y(sized)30 b(circuits)f(\(regarded)h(on)h(a)f(par)g(with)g(pro)m (ving)f(P)c Fo(6)p Fr(=)g(NP\))31 b(m)m(ust)f(surmoun)m(t)g(either)g (the)g(largeness)g(or)h(the)0 430 y(lo)m(w-complexit)m(y)f(condition.) 141 543 y(Although)g(the)h(Razb)s(oro)m(v-Rudic)m(h)f(framew)m(ork)h (and)f(results)f(ha)m(v)m(e)j(not)f(y)m(et)h(b)s(een)e(carried)f(o)m(v) m(er)j(formally)0 656 y(to)38 b(arithmetic)f(circuits,)h(it)e(is)h (reasonable)g(to)h(sp)s(eak)f(as)h(though)f(they)g(ha)m(v)m(e|and)h(to) g(exp)s(ect)g(that)g(recen)m(t)0 769 y(tec)m(hniques)d(b)m(y)h(Koiran)f ([Koi96])h(and)f(B)s(\177)-48 b(urgisser)35 b([B)s(\177)-48 b(ur98)q(,)36 b(B)s(\177)-48 b(ur00])36 b(using)e(\014elds)h(of)g (\014nite)g(c)m(haracteristic)i(as)0 882 y(conduits)27 b(from)h(the)g(algebraic)g(to)h(the)f(Bo)s(olean)g(case)i(can)e(extend) g(to)h(accomplish)e(this.)39 b(Th)m(us)27 b(assuming)g(the)0 995 y(Mulm)m(uley-Sohoni)e(metho)s(d)i(is)f(useful)g(against)i(\(the)g (arithmetical)f(analogue)h(of)7 b(\))28 b(P)o Fj(=)p Fi(p)-5 b(oly)11 b Fr(,)28 b(whic)m(h)e(of)i(\(b\))g(and)0 1108 y(\(c\))j(do)s(es)f(it)g(escap)s(e)h(from?)141 1220 y(Mulm)m(uley)g(and)h(Sohoni)f(argue)h(for)g(\(b\).)47 b(Indeed,)33 b(their)e(metho)s(dology)h(for)g(the)h(case)g(of)g(the)g (p)s(ermanen)m(t)0 1333 y(and)26 b(determinan)m(t)h(as)g(outlined)e(ab) s(o)m(v)m(e)j(is)e(highly)e(sp)s(eci\014c)i(to)i(them|and)e(w)m(ould)f (not)j(pro)m(v)m(e)f(the)g(hardness)f(of)0 1446 y(a)32 b(\\random")f(function)f(as)h(do)g(natural)f(pro)s(ofs.)43 b(Note)32 b(that)g(if)e(the)i(hardness)d(predicate)i(\005)3227 1460 y Fe(n)3275 1446 y Fr(\()p Fj(h)p Fr(\))h(w)m(ere)f(simply)0 1559 y Fj(D)75 1573 y Fe(n)122 1559 y Fr(\()p Fj(h)p Fr(\))c(=)c(\\)p Fj(h)h Fr(do)s(es)f(not)h(b)s(elong)f(to)h(the)g(pro)5 b(jectiv)m(e)24 b(Z-closure)f(of)g(the)h(orbit)f(of)h(the)f(determinan) m(t,")j(then)d(\005)3730 1573 y Fe(n)3777 1559 y Fr(\()p Fj(h)p Fr(\))0 1672 y(w)m(ould)30 b(b)s(e)h(v)m(ery)h(large|indeed)f Fi(vast)41 b Fr(in)30 b(the)i(Zariski)d(sense,)j(b)s(eing)f(the)g (complemen)m(t)h(of)g(the)g(closure)f(of)h(the)0 1785 y(orbit.)40 b(Ho)m(w)m(ev)m(er,)32 b(they)e(emphasize)f(that)h(their)f (hardness)g(predicates)g(will)e(ha)m(v)m(e)k(the)f(form)f Fj(D)3329 1799 y Fe(n)3377 1785 y Fr(\()p Fj(h)p Fr(\))45 b Fo(^)f Fj(S)3705 1799 y Fe(n)3752 1785 y Fr(\()p Fj(h)p Fr(\),)0 1898 y(or)31 b(more)g(lik)m(ely)f Fj(D)656 1912 y Fe(n)703 1898 y Fr(\()p Fj(h)p Fr(\))48 b Fo(^)f Fj(P)1039 1912 y Fe(n)1086 1898 y Fr(\()p Fj(h)p Fr(\),)32 b(where)f Fj(S)1585 1912 y Fe(n)1663 1898 y Fr([)p Fj(P)1746 1912 y Fe(n)1793 1898 y Fr(])g(expresses)g(the)g([partial])g(stabilit)m(y)e (of)j Fj(h)f Fr(under)e(the)j(same)0 2011 y(group)39 b(action|plus)f(the)i(abilit)m(y)f(to)h(compute)g(explicit)e (obstructions.)68 b(The)39 b(predicates)h Fj(S)3361 2025 y Fe(n)3447 2011 y Fr(and)g Fj(P)3692 2025 y Fe(n)3779 2011 y Fr(are)0 2124 y(exp)s(ected)27 b(to)f(b)s(e)g(small)f(\(under)g (translation)g(to)i(the)f(Bo)s(olean)g(case\).)41 b(Moreo)m(v)m(er,)30 b(for)25 b Fj(h)i Fr(suc)m(h)f(as)g(the)g(\(padded\))0 2237 y(p)s(ermanen)m(t)h(p)s(olynomial\(s\),)g(one)h(can)g(conjoin)f (to)h Fj(S)1857 2251 y Fe(n)1932 2237 y Fr(or)f Fj(P)2098 2251 y Fe(n)2173 2237 y Fr(the)h(clause,)h(\\and)e Fj(h)h Fr(has)g(a)g(nice,)g(large)g(isotrop)m(y)0 2350 y(subgroup)h(suc)m(h)h (that)h Fj(h)f Fr(is)g(the)g(only)g(thing)f(it)h(\014xes.")141 2462 y(Ho)m(w)m(ev)m(er,)37 b(in)c(public)e(talks)j(subsequen)m(t)f(to) i(their)e(pap)s(er,)h(Razb)s(oro)m(v)h(and)e(Rudic)m(h)f(ha)m(v)m(e)k (p)s(oin)m(ted)c(out)j(a)0 2575 y(philosophical)30 b(obstacle)j(to)h (argumen)m(ts)g(that)f(rely)g(only)f(on)h(o)m(v)m(ercoming)h(\(b\).)49 b(Within)32 b(the)h(con\014nes)f(of)i(the)0 2688 y(purp)s(orted)28 b(pro)s(of)h(based)g(on)h(\005)1110 2702 y Fe(n)1157 2688 y Fr(,)g(the)g(complemen)m(t)g(\005)1943 2655 y Ff(0)1943 2711 y Fe(n)2020 2688 y Fr(b)s(ecomes)g(an)g(\\easyness")h (predicate.)40 b(If)29 b(\005)3547 2655 y Ff(0)3547 2711 y Fe(n)3624 2688 y Fr(is)g(v)-5 b(ast,)0 2801 y(then)24 b(one)h(is)e(willy-nilly)d(arguing)j(in)g(the)i(situation)e(where)h (\\a)i(random)d(function)g Fj(f)34 b Fr(is)23 b(easy)-8 b(.")41 b(No)m(w)25 b(b)s(ecause)f(+)0 2914 y(is)e(in)m(v)m(ertible)g (\(as)i(with)e(exclusiv)m(e-or)h(in)f(the)h(Bo)s(olean)h(case\),)i(it)d (follo)m(ws)f(that)i(\\a)g(random)f(function)e Fj(g)29 b Fr(=)c Fj(h)6 b Fo(\000)g Fj(f)k Fr(")0 3027 y(o)m(v)m(er)38 b(random)d Fj(f)46 b Fr(is)35 b(easy)-8 b(.)60 b(One)36 b(th)m(us)g(m)m(ust)g(admit)g(cases)h(where)f(one's)g(\\hard")h (function)e Fj(h)h Fr(equals)g(a)h(sum)0 3140 y Fj(f)e Fr(+)25 b Fj(g)42 b Fr(of)c(t)m(w)m(o)i(\\easy")f(functions.)63 b(Roughly)38 b(put,)h(this)e(prev)m(en)m(ts)i(the)g(lo)m(w)m(er-b)s (ound)d(pro)s(of)i(from)g(w)m(orking)0 3253 y(inductiv)m(ely)28 b(on)i(the)h(arithmetical)e(op)s(erators.)141 3366 y(Ho)m(w)m(ev)m(er,) 47 b(the)41 b(argumen)m(t)h(o)m(v)m(er)h(\(b\))f(ma)m(y)g(b)s(e)f (e\013ectiv)m(ely)h(mo)s(ot,)k(as)41 b(it)g(seems)h(transparen)m(t)g (from)f(the)0 3479 y(w)m(a)m(y)35 b(the)e(Mulm)m(uley-Sohoni)e(tec)m (hnique)j(in)m(v)m(olv)m(es)f(\\large)i(ob)5 b(jects")34 b(that)h(the)e(complexit)m(y)h(of)f(the)h(hardness)0 3592 y(predicate)26 b(w)m(ould)e(b)s(e)i(greater)h(than)f(the)g(b)s (ound)e(in)g(\(c\).)40 b(The)26 b(large)g(ob)5 b(jects)27 b(ha)m(v)m(e)g(size)f(2)3112 3559 y Fe(p)p Fm(\()p Fe(n)p Fm(\))3275 3592 y Fr(where)g Fj(p)f Fr(is)g Fi(not)36 b Fr(a)0 3704 y(\014xed)26 b(p)s(olynomial)f(but)h(rather)h(one)h(that) f(is)f(univ)m(ersally)f(quan)m(ti\014ed.)39 b(F)-8 b(or)27 b(instance,)h(the)f(hardness)f(predicate)0 3817 y(tak)m(en)37 b(from)f(their)f(\\Conjecture)h(4.3")i(for)e(the)g(p)s(ermanen)m(t)f (could)h(b)s(e)f(made)h(to)h(read:)52 b Fj(h)36 b Fr(is)f(hard)g(if)g (for)h(all)0 3930 y(p)s(olynomials)d Fj(p)p Fr(\()p Fj(n)p Fr(\),)38 b(taking)e Fj(m)e Fr(=)g Fj(p)p Fr(\()p Fj(n)p Fr(\),)k Fj(h)e Fr(padded)f(up)g(to)h Fj(h)2201 3897 y Fe(\036)2284 3930 y Fr(is)f(not)h(in)f(the)h(orbit)f(closure)g(of)h (det)3594 3944 y Fe(m)3661 3930 y Fr(.)57 b(The)0 4043 y(\\Natural)28 b(Pro)s(ofs)f(Obstacle,")i(ho)m(w)m(ev)m(er,)h(still)c (seems)i(to)h(indicate)e(here)g(that)i(their)e(metho)s(d)g(will)e(not)j (b)s(e)f(able)0 4156 y(to)k(escap)s(e)f(the)h(necessit)m(y)f(of)h(quan) m(tifying)d(o)m(v)m(er)k Fj(p)d Fr(and)h(dealing)f(b)s(oth)g(with)g (the)h(padding)f(and)g(with)g Fi(multiple)0 4269 y Fr(source)i (functions)d(lik)m(e)i(det)963 4283 y Fe(m)1060 4269 y Fr(that)h(one)g(has)f(to)h(argue)g(o)m(v)m(er.)141 4382 y(Mulm)m(uley)36 b(and)h(Sohoni)f(do)i(ha)m(v)m(e)g(a)g(p)s(oin)m (t)f(in)f(giving)h(motiv)-5 b(ation)37 b(to)h(consider)f(problems)e (other)j(than)0 4495 y(standard)f(\(NP-\)complete)i(ones)e(as)h (targets)h(for)e(lo)m(w)m(er)h(b)s(ound)e(pro)s(ofs.)61 b(In)37 b(Sections)g(7)h(and)f(8)h(of)f([MS02)q(],)0 4608 y(they)31 b(in)m(tro)s(duce)e(the)h(follo)m(wing)f(problem)g(in)g (NP,)i(whic)m(h)e(w)m(e)i(name)f(\\)p Fb(Full)k(Rank)f(A)-11 b(v)n(oid)n(ance)p Fr(.")141 4811 y Fb(Inst)-6 b(ance:)50 b Fr(A)31 b(matrix)e Fj(X)38 b Fr(with)29 b Fj(n)h Fr(ro)m(ws)g(and)g Fj(k)s(n)g Fr(columns)f(group)s(ed)g(as)i Fj(n)f Fr(blo)s(c)m(ks)g(of)g Fj(k)s Fr(.)141 4924 y Fb(Question:)f Fr(Is)39 b(it)g(p)s(ossible)d(to) k(c)m(ho)s(ose)h(one)e(column)f(from)h(eac)m(h)h(blo)s(c)m(k)f(so)h (that)f(the)h(resulting)620 5037 y Fj(n)20 b Fo(\002)g Fj(n)30 b Fr(matrix)f Fj(M)41 b Fr(has)30 b(det\()p Fj(M)10 b Fr(\))26 b(=)f(0?)0 5227 y(They)h(conjecture)i(that)g(ev)m(en)g(for)e Fj(k)j Fr(=)c(3)i(and)g Fj(F)38 b Fr(=)25 b(GF)q(\(2\),)k(this)d (problem)f(is)h(not)h(in)f(P.)39 b(The)27 b(closest)h(problem)0 5340 y(that)h(is)d(kno)m(wn)i(to)h(us)e(\(p)s(ersonal)g(comm)m (unication)g(from)g(Mitsunori)f(Ogihara,)i(Septem)m(b)s(er)g(2002\))i (to)e(b)s(e)g(NP-)p eop %%Page: 13 13 13 12 bop 0 91 a Fr(complete)34 b(has)e Fj(k)h Fr(=)d(2)j(and)g(2)p Fj(r)j Fr(columns)c(where)g(the)i(input)d(can)i(v)-5 b(ary)33 b Fj(r)j Fr(sub)5 b(ject)33 b(to)h Fj(r)e(<)d(n)p Fr(,)34 b(and)e(asks)i(if)e(the)0 204 y(resulting)j Fj(n)24 b Fo(\002)g Fj(r)40 b Fr(matrix)c Fj(M)47 b Fr(has)36 b(column)g(rank)g(less)g(than)h Fj(r)s Fr(.)60 b(The)36 b(Mulm)m(uley-Sohoni)e(problem)h(requires)0 317 y Fj(r)28 b Fr(=)d Fj(n)p Fr(,)i(ho)m(w)m(ev)m(er,)j(and)d(Ogihara's)f(reduction) g(\(from)i(3-not-all-equal-SA)-8 b(T,)27 b(and)g(adaptable)g(for)g(an)m (y)h(\014eld)e Fj(F)13 b Fr(\))0 430 y(seems)31 b(to)g(require)e Fj(r)f(<)d(n)k Fr(dep)s(ending)f(on)j(the)f(input)f(form)m(ula)g(to)i (the)g(reduction.)141 543 y(Mulm)m(uley)h(and)h(Sohoni)g(pro)m(v)m(e)h (that)h(pro)m(vided)d(the)i(c)m(haracteristic)g(of)g(the)g(\014eld)e Fj(F)47 b Fr(do)s(es)34 b(not)g(divide)d Fj(n)p Fr(,)0 656 y Fj(k)s Fr(,)e(or)f Fj(k)20 b Fo(\000)15 b Fr(1,)30 b(the)e(follo)m(wing)f(p)s(olynomial)f Fj(E)5 b Fr(\()p Fj(X)i Fr(\))29 b(asso)s(ciated)g(to)g(this)e(problem)g(is)g(stable)i (under)d(the)j(standard)0 769 y(action)i(b)m(y)f Fi(SL)507 788 y Fe(k)r(n)589 770 y Fh(2)627 769 y Fr(\()p Fj(F)14 b Fr(\))30 b(\(again)h(\\unrolling")d(the)j(matrix)e(in)m(to)i(a)f(v)m (ector\):)1544 963 y Fj(E)5 b Fr(\()p Fj(X)i Fr(\))27 b(=)1891 883 y Fa(Y)1922 1057 y Fe(\033)2012 963 y Fr(det\()p Fj(X)2248 977 y Fe(\033)2296 963 y Fr(\))p Fj(;)0 1218 y Fr(where)33 b Fj(\033)j Fr(stands)d(for)g(functions)f(c)m(ho)s(osing) h(one)h(column)e(from)h(eac)m(h)h(blo)s(c)m(k)f(and)g Fj(X)2964 1232 y Fe(\033)3044 1218 y Fr(is)g(the)g(resulting)f Fj(n)22 b Fo(\002)f Fj(n)0 1331 y Fr(matrix.)47 b(Clearly)31 b Fj(E)5 b Fr(\()p Fj(X)i Fr(\))30 b(=)f(0)k(i\013)f(the)g(answ)m(er)h (to)g(the)g(problem)e(instance)h(is)g(\\y)m(es.")48 b(The)33 b(stabilit)m(y)e(of)h Fj(E)5 b Fr(\()p Fj(X)i Fr(\))0 1444 y(then)32 b(pla)m(ys)f(in)m(to)h(partial)f(stabilit)m(y)f(of)i (padded)f(forms)g Fj(E)5 b Fr(\()p Fj(X)i Fr(\))2180 1411 y Fe(\036)2180 1467 y(n)2229 1444 y Fr(,)32 b(and)g(th)m(us)f (allo)m(ws)g(the)h(meat)h(of)f(the)g(analysis)0 1557 y(to)f(come)g(in)e([MS02b)q(])i(to)g(b)s(e)e(applied)g(to)i(it.)141 1670 y(In)d(conclusion,)f(there)i(is)e(b)s(oth)h(m)m(uc)m(h)g(deep)h (mathematical)f(con)m(ten)m(t)j Fi(and)38 b Fr(some)29 b(new)f(concrete)i(com)m(bina-)0 1783 y(torics)i(in)f(their)h(approac)m (h.)47 b(It)32 b(ma)m(y)h(not)g(seem)g(near)f(to)h(resolving)e(P)h(vs.) h(NP)f(no)m(w,)h(but)f(it)g(do)s(es)g(talk)g(ab)s(out)0 1896 y(ob)5 b(jects)31 b(of)g(the)f(righ)m(t)g(kind)f(of)h(complexit)m (y)g(for)h(w)m(orking)e(on)h(it.)0 2135 y Fl(Ac)m(kno)m(wledgmen)m(ts) 91 b Fr(The)54 b(author)h(thanks)f(Ketan)h(Mulm)m(uley)-8 b(,)60 b(Mitsunori)53 b(Ogihara,)60 b(and)54 b(Maurice)0 2247 y(Jansen)30 b(for)g(helpful)d(answ)m(ers)k(and)e(con)m (tributions.)0 2532 y Fk(References)0 2735 y Fr([AB95])124 b(J.L.)32 b(Alp)s(erin)d(and)i(R.B.)i(Bell.)43 b Fi(Gr)-5 b(oups)36 b(and)f(R)-5 b(epr)g(esentations)p Fr(,)35 b(v)m(olume)d(162)h(of)f Fi(Gr)-5 b(aduate)36 b(T)-7 b(exts)396 2848 y(in)33 b(Mathematics)p Fr(.)42 b(Springer)28 b(V)-8 b(erlag,)31 b(1995.)0 3033 y([BCS97])75 b(P)-8 b(.)31 b(B)s(\177)-48 b(urgisser,)28 b(M.)j(Clausen,)d(and)h(M.A.)i (Shokrollahi.)36 b Fi(A)n(lgebr)-5 b(aic)32 b(Complexity)h(The)-5 b(ory)p Fr(.)41 b(Springer)396 3146 y(V)-8 b(erlag,)32 b(1997.)0 3330 y([B)s(\177)-48 b(ur98])105 b(P)m(eter)34 b(B)s(\177)-48 b(urgisser.)44 b(On)31 b(the)h(structure)g(of)g(V)-8 b(alian)m(t's)33 b(complexit)m(y)e(classes.)46 b(In)31 b Fi(15th)36 b(A)n(nnual)e(Sym-)396 3443 y(p)-5 b(osium)31 b(on)f(The)-5 b(or)g(etic)g(al)32 b(Asp)-5 b(e)g(cts)31 b(of)f(Computer)g(Scienc)-5 b(e)p Fr(,)28 b(v)m(olume)f(1373)i(of)e Fi(lncs)p Fr(,)h(pages)g(194{204,)396 3556 y(P)m(aris)i(F)-8 b(rance,)32 b(25{27)h(F)-8 b(ebruary)30 b(1998.)i(Springer.)0 3740 y([B)s(\177)-48 b(ur00])105 b(P)m(eter)27 b(B)s(\177)-48 b(urgisser.)31 b(Co)s(ok's)25 b(v)m(ersus)g(Valian)m(t's)g(h)m(yp)s (othesis.)31 b Fi(The)-5 b(or.)29 b(Comp.)g(Sci.)p Fr(,)e(235:71{88,)j (2000.)0 3925 y([Hum81])61 b(J.E.)26 b(Humphreys.)32 b Fi(Line)-5 b(ar)29 b(A)n(lgebr)-5 b(aic)28 b(Gr)-5 b(oups)p Fr(,)29 b(v)m(olume)d(21)h(of)f Fi(Gr)-5 b(aduate)30 b(T)-7 b(exts)29 b(in)f(Mathematics)p Fr(.)396 4038 y(Springer)h(V)-8 b(erlag,)31 b(1981.)42 b(2nd)30 b(prin)m(ting.)0 4222 y([Koi96])114 b(P)m(ascal)31 b(Koiran.)38 b(Hilb)s(ert's)28 b(Nullstellensatz)g(is)g(in)g(the)i(p)s(olynomial)d(hierarc)m(h)m(y)-8 b(.)40 b Fi(Journal)32 b(of)h(Com-)396 4335 y(plexity)p Fr(,)f(12\(4\):273{286,)k(Decem)m(b)s(er)31 b(1996.)0 4519 y([Lok95])106 b(S.)30 b(Lok)-5 b(am.)40 b(Sp)s(ectral)29 b(metho)s(ds)g(for)h(matrix)f(rigidit)m(y)f(with)g(applications)g(to)j (size-depth)e(tradeo\013s)396 4632 y(and)g(comm)m(unication)f (complexit)m(y)-8 b(.)38 b(In)28 b Fi(Pr)-5 b(o)g(c.)32 b(36th)h(A)n(nnual)e(IEEE)f(Symp)-5 b(osium)33 b(on)f(F)-7 b(oundations)396 4745 y(of)33 b(Computer)h(Scienc)-5 b(e)p Fr(,)31 b(pages)g(6{15,)h(1995.)0 4930 y([MS02])122 b(K.)32 b(Mulm)m(uley)d(and)i(M.)h(Sohoni.)42 b(Geometric)32 b(complexit)m(y)g(theory)f(I:)h(An)e(approac)m(h)i(to)g(the)g(P)f(vs.) 396 5043 y(NP)g(and)f(related)g(problems.)39 b Fi(SIAM)32 b(J.)g(Comput.)p Fr(,)f(31\(2\):496{526,)36 b(2002.)0 5227 y([MS02b])71 b(K.)42 b(Mulm)m(uley)e(and)h(M.)i(Sohoni.)73 b(Geometric)42 b(complexit)m(y)g(theory)g(I)s(I:)f(Explicit)f (obstructions.)396 5340 y(Man)m(uscript,)30 b(in)f(preparation,)h (2001{02.)p eop %%Page: 14 14 14 13 bop 0 91 a Fr([MS02c])82 b(K.)34 b(Mulm)m(uley)f(and)g(M.)i (Sohoni.)50 b(Geometric)34 b(complexit)m(y)g(theory)-8 b(,)36 b(P)e(vs.)g(NP,)g(and)f(explicit)g(ob-)396 204 y(structions.)28 b(T)-8 b(o)24 b(app)s(ear)e(in)g Fi(Pr)-5 b(o)g(c)g(e)g(e)g(dings,)29 b(International)f(Confer)-5 b(enc)g(e)27 b(on)g(A)n(lgebr)-5 b(a)25 b(and)i(Ge)-5 b(ometry,)396 317 y(Hyder)g(ab)g(ad,)35 b(2001)p Fr(,)e(refereed)d(pro) s(ceedings)f(forthcoming.)0 505 y([Mul99])96 b(K.)21 b(Mulm)m(uley)-8 b(.)24 b(Lo)m(w)m(er)d(b)s(ounds)e(in)g(a)i(parallel)e (mo)s(del)h(without)f(bit)h(op)s(erations.)k Fi(SIAM)f(J.)h(Comput.)p Fr(,)396 618 y(28:1460{1509,)36 b(1999.)0 805 y([Mum76])46 b(D.)55 b(Mumford.)111 b Fi(A)n(lgebr)-5 b(aic)54 b(Ge)-5 b(ometry)56 b(I:)e(Complex)i(Pr)-5 b(oje)g(ctive)55 b(V)-7 b(arieties)p Fr(,)60 b(v)m(olume)54 b(221)i(of)396 918 y Fi(Grund)5 b(lehr)-5 b(en)35 b(der)e(mathematischen)i(Wissenschaften) p Fr(.)41 b(Springer)29 b(V)-8 b(erlag,)31 b(1976.)0 1106 y([PV91])126 b(V.)46 b(P)m(op)s(o)m(v)g(and)f(E.)h(Vin)m(b)s(erg.) 84 b Fi(Invariant)48 b(The)-5 b(ory)p Fr(,)51 b(v)m(olume)45 b(55)i(of)e Fi(A)n(lgebr)-5 b(aic)46 b(Ge)-5 b(ometry)48 b(IV,)396 1219 y(Encyclop)-5 b(e)g(dia)35 b(Math.)e(Sci.)40 b Fr(Springer)28 b(V)-8 b(erlag,)32 b(1991.)0 1406 y([RR97])122 b(A.)31 b(Razb)s(oro)m(v)g(and)f(S.)g(Rudic)m(h.)39 b(Natural)30 b(pro)s(ofs.)40 b Fi(J.)32 b(Comp.)i(Sys.)f(Sci.)p Fr(,)d(55:24{35,)k (1997.)0 1594 y([V)-8 b(al79])125 b(L.)45 b(V)-8 b(alian)m(t.)84 b(Completeness)44 b(classes)g(in)f(algebra.)84 b(T)-8 b(ec)m(hnical)44 b(Rep)s(ort)g(CSR-40-79,)50 b(Dept.)c(of)396 1707 y(Computer)30 b(Science,)g(Univ)m(ersit)m(y)g(of)h(Edin)m(burgh,)c (April)h(1979.)0 1895 y([vzG87])97 b(J.)35 b(v)m(on)f(zur)g(Gathen.)53 b(F)-8 b(easible)34 b(arithmetic)f(computations:)49 b(V)-8 b(alian)m(t's)35 b(h)m(yp)s(othesis.)50 b Fi(Journal)38 b(of)396 2007 y(Symb)-5 b(olic)34 b(Computation)p Fr(,)f(4:137{172,)i (1987.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF