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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 73 /I put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 48 /prime put dup 50 /element put dup 54 /negationslash put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 100 /d put dup 105 /i put dup 107 /k put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 70 /F put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 58 /period put dup 59 /comma put dup 65 /A put dup 67 /C put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 EABE595BC964156D9D8C2BAB0F49664E951D7C1A3D1789C47F03C7051A63D5E8 DF04FAAC47351E82CAE0794AA9692C6452688A74A7A6A7AD09B8A9783C235EC1 EA2156261B8FB331827145DE315B6EC1B3D8B67B3323F761EAF4C223BB214C4C 6B062D1B281F5041D068319F4911058376D8EFBA59884BA3318C5BC95684F281 E0591BC0D1B2A4592A137FF301610019B8AC46AE6E48BC091E888E4487688350 E9AD5074EE4848271CE4ACC38D8CBC8F3DB32813DDD5B341AF9A6601281ABA38 4A978B98483A63FCC458D0E3BCE6FD830E7E09B0DB987A6B63B74638FC9F21A5 8C68479E1A85225670D79CDDE5AC0B77F5A994CA700B5F0FF1F97FC63EFDE023 8135F04A9D20C31998B12AE06676C362141AAAA395CDEF0A49E0141D335965F2 FB4198499799CECCC8AA5D255264784CD30A3E8295888EFBC2060ADDD7BAC45A EEEECDFF7A47A88E69D84C9E572616C1AC69A34B5F0D0DE8EE4EDF9F4ADE0387 680924D8D5B73EF04EAD7F45977CA8AD73D4DD45DE1966A3B8251C0386164C35 5880DD2609C80E96D1AB861C9259748E98F6711D4E241A269ED51FF328344664 3AF9F18DCE671611DB2F5D3EA77EE734D2BED623F973E6840B8DAD1E2C3C2666 DD4DD1C1CBB6B4446309621BA371A521AA5D66E5A9ABE07A801317469B675AA2 C0DDCEDF30C74D4A1BE90C14058E8086338EA5CFA576C0424F4B8A5AC241A710 E75F6145B3EC6A7C9BF891A817046148F8C19DF03CA12126A7C57579EEE81E24 D6E38F8008D1F770C91822A01F6A638438D6E22B2251669C1F504530651446CB C312AED93197B356CD4008451E325A5D4799D005D49B3CBD1C51B107F43628ED 0698870060D44738B1E4E326AF534B3A5B47AFC98B289D0BB06A87B76615CC18 D31A886F467E8566B7F6B6CDAF4C5C66401AB6B046763297F332195C071F6BC9 CD6D291A63A91DEA0656C3EF8F75FEBA07224DED3BA13F87AD4D226FCF2479F9 905212B281B6F27584B08CA42D1E38651A58281129B0609702BBEDA93CB85406 D5AF07486B8326AA7D8E7431FF12646F776B3A56E33B3F55C09772A7F307ED3A 223318DAC61993CC48049D924E1D940A2589632EBEF98BBD9242D2E8CB1BFE22 E83B79502725AD928123F67F8EC2A1F7DD029919ABE31CB4ED85C2856E2046F7 9A4D63F9995AD55FA45A983C4E9DC967BEF4D6F973AFA5EDB696A43873B052CF 1D85EF3E3F51D560AC14F69D79D8DCE3A63BEB8C8A21C0F268124B066B63D730 85816E7067DD09EA2E6807FE6D8DD88AC2A31DD0B24BC319C1E57A4C0DEEC271 D37CDE08CF72D3550AA7A2A13D8A844951C0C66CEF0A4CA7B9AFFF52DF677558 559783CC0493B3ABDE1BB7725AC81F0B4204D483B8DFA9FA8444A2732F6E7FED C9485C2EB1C465D624D300385B3A8706928BF6DA0ED85BB12D3EEA33CB655C73 8731CFFAACF31D79BBC0AA0A42F178D52F5AD75E663790D69D2D9BD0575A32EE A81CD6CF14005FCC03FB13B8F0AC2AE7F7E0190C4F8045A10D0E229DE768BF7C 2E5C168A035E33CC771260CCCCC51BD67C7C55821A9FEC9C8F7B2558B84CFEE5 A22DFE43B4B1F00F8B1FE07F166EE0539D143553C96053ECF9AF1388811E8BE7 B8BEB72EE83A0456B7C23BD866C2ED4F403869756EDD832F4A8B1EC0E3B12FDB AF35554E009072E4A6EB1A75A814E32ECF30F9FE856F78EF66E2B369BCCD6E08 533DE0D46387A31978E1C3113EAD9F79A663DE445FCA43E65315147D2D56C95E D9F08A660A7F1E775F2E14912125CD8C90EC9348E9708E7AD37DECA493B88E9C 5095A11A8E491F83A2D47E13512F5AFAA993F4D2D6A7A1833B6D2BE8312B4E98 46CB9CBEFFB1BF235D25C22B43B718758565317C7FA98EEC1642409ED498478F 95A7C8037D0CF1ABCCC8289C6887C43909A907C347F9805DB7261E14B3ECAF75 A96ED7F234E2B0C5A7B4FF4C08E417E4502729F559C883F427C92DE4EC9F06CD 5372BFBBCD365768D5B2A2B9B6FEB88F1A079F6BF58087CC3067C51A18B91457 6690C24A1AC265E20310CFE84DB39F46B0C879F25CC89B4020EBEF2560FB684A 292215F4B89E0C61B45B491EEC14E42B3DBD2C04321F7FC885FC31B056C2B679 86C326D570461E83D883CD1263429DFC1DC4AEE1C80AB2B4B90199F83F79187A DBC53B911FFEB1084ED04109F276646AE245AF7A670266674A7F271683D9EDF7 1D4D6E9EC1EE98C878BE0A5C586EAE5C5C5AD55F278842980531DBCA828FC32B E37C7C6201909A8A56A1A5CDF159AB4E4498112C5CB9E370D55987BA70FD3401 DBF6E4DAE2D5E78CED7F03BBC9369395117F93BD8C2EE18644A441AA3C63C36C FA0C6A44D9683B6BD6BBF42A2A88727E1A98CE57A2A42FDC81C95B4018FF5524 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 103 /g put dup 108 /l put dup 111 /o put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 73 /I put dup 76 /L put dup 80 /P put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 116 /unionsq put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 25 /pi put dup 28 /tau put dup 30 /phi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fn(Lemma)39 b(5.2.)p Black 42 w Fw(L)-5 b(et)36 b Fu(W)1133 1759 y Fs(1)1173 1744 y Fu(;)17 b(:)g(:)g(:)f(;)h(W)1484 1759 y Fq(s)1550 1744 y Ft(\022)30 b Fm(F)1718 1708 y Fq(n)1801 1744 y Fw(b)-5 b(e)35 b(\034xe)-5 b(d)35 b(subsp)-5 b(ac)g(es,)35 b(e)-5 b(ach)35 b(of)h(dimension)e(at)257 1864 y(most)h Fu(t)p Fw(.)45 b(Consider)33 b(the)i(line)-5 b(ar)34 b(tr)-5 b(ansformation)34 b(\(for)h(some)f Fu(\013)28 b Ft(2)g Fm(F)p Fw(\),)737 2087 y Fu(\036)795 2102 y Fq(\013;t;n)952 2087 y Fv(\()p Fu(x)1045 2102 y Fs(1)1085 2087 y Fu(;)17 b(:)g(:)g(:)e(;)i(x)1358 2102 y Fq(n)1406 2087 y Fv(\))27 b(:=)h(\(\()p Fu(\013)1741 2046 y Fq(i)p Fs(\()p Fq(j)t Fj(\000)p Fs(1\))1946 2087 y Fv(\)\))2022 2102 y Fs(1)p Fj(\024)p Fq(i)p Fj(\024)p Fq(t;)p Fs(1)p Fj(\024)p Fq(j)t Fj(\024)p Fq(n)2483 2087 y Ft(\001)21 b Fv([)p Fu(x)2614 2102 y Fs(1)2671 2087 y Ft(\001)c(\001)g(\001)e Fu(x)2859 2102 y Fq(n)2906 2087 y Fv(])2933 2046 y Fq(T)2988 2087 y Fu(:)257 2309 y Fw(Then)30 b(ther)-5 b(e)30 b(ar)-5 b(e)30 b(at)g(most)g Fu(snt)1389 2273 y Fs(2)1459 2309 y Fw(elements)f Fu(\013)f Ft(2)g Fm(F)j Fw(for)e(which)h(the)g (dimension)e(of)i(some)257 2429 y Fu(W)349 2444 y Fq(i)413 2429 y Fw(dr)-5 b(ops,)34 b(i.e.)44 b Fu(dim)p Fv(\()p Fu(\036)1147 2444 y Fq(\013;t;n)1304 2429 y Fv(\()p Fu(W)1434 2444 y Fq(i)1462 2429 y Fv(\)\))28 b Fu(<)f(dim)p Fv(\()p Fu(W)1968 2444 y Fq(i)1997 2429 y Fv(\))p Fw(.)p Black 257 2637 a(Pr)-5 b(o)g(of.)p Black 49 w Fx(The)28 b(pro)s(of)e(idea)g (is)h(to)g(capture)h(all)d(the)i(\020bad\021)34 b Fu(\013)q Fx('s)28 b(in)e(a)h(univ)-5 b(ariate)25 b(equation,)257 2757 y(then)34 b(its)e(degree)h(upp)s(er)g(b)s(ounds)g(their)f(n)m(um)m (b)s(er.)404 2878 y(Whenev)m(er)i(the)f(dimension)e(of)i(the)f(image)f (of)h Fu(W)2254 2893 y Fq(i)2315 2878 y Fx(under)h Fu(\036)2649 2893 y Fq(\013;t;n)2839 2878 y Fx(drops,)g(it)e(means)257 2999 y(that)26 b(the)g(top-left)e Fu(dim)p Fv(\()p Fu(W)1259 3014 y Fq(i)1288 2999 y Fv(\))8 b Ft(\002)g Fu(dim)p Fv(\()p Fu(W)1718 3014 y Fq(i)1747 2999 y Fv(\))26 b Fx(submatrix)f(of)g(\(the)h(matrix)e(de\034ning\))i Fu(\036)3339 3014 y Fq(\013;t;n)257 3119 y Fx(is)d(singular.)38 b(Th)m(us,)26 b(its)d(determinan)m(t)f(giv)m(es)h(us)g(a)f(univ)-5 b(ariate)21 b(equation)i(in)f Fu(\013)h Fx(of)f(degree)257 3239 y(at)29 b(most)g Fu(nt)702 3203 y Fs(2)742 3239 y Fx(.)42 b(Doing)29 b(this)g(for)f(all)g Fu(W)1651 3254 y Fq(i)1679 3239 y Fx('s)i(giv)m(es)f(us)h(the)g(promised)f(b)s(ound)g (of)g Fu(snt)3255 3203 y Fs(2)3295 3239 y Fx(.)p 3425 3239 4 66 v 3429 3177 59 4 v 3429 3239 V 3487 3239 4 66 v 404 3441 a(No)m(w,)47 b(follo)m(wing)42 b([)p Fn(?)p Fx(],)47 b(w)m(e)e(giv)m(e)f(the)h(construction)f(of)g(the)g(subspaces) j Fu(W)3241 3456 y Fq(i)3269 3441 y Fx('s)e(for)257 3562 y(the)40 b(case)g(of)f Fv(\006\005\006\()p Fu(n;)17 b(k)s(;)g(d)p Fv(\))39 b Fx(circuits)g(assuming)f(a)h(rank)h(b)s(ound)f(of)g Fu(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))38 b Fx(for)h(the)257 3682 y(minimal)29 b(simple)i(iden)m(tities.)257 3701 y SDict begin H.S end 257 3701 a 257 3701 a SDict begin 14.5 H.A end 257 3701 a 257 3701 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.3) cvn H.B /DEST pdfmark end 257 3701 a Black 188 x Fn(Theorem)k(5.3.)p Black 40 w Fw(L)-5 b(et)33 b Fu(C)40 b Fw(b)-5 b(e)33 b(a)f Fv(\006\005\006\()p Fu(n;)17 b(k)s(;)g(d)p Fv(\))33 b Fw(cir)-5 b(cuit)33 b(and)g Fu(S)g Ft(\022)28 b Fm(F)33 b Fw(b)-5 b(e)33 b(of)f(size)g(at)h(le)-5 b(ast)257 4010 y Fu(n)p Fv(2)364 3974 y Fq(k)407 4010 y Fu(d)458 3974 y Fs(2)497 4010 y Fu(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))797 3974 y Fs(2)836 4010 y Fw(.)67 b(If)41 b Fu(C)49 b Fw(is)42 b(a)g(nonzer)-5 b(o)41 b(cir)-5 b(cuit)43 b(then)f(ther)-5 b(e)42 b(is)g(an)f Fu(\013)h Ft(2)g Fu(S)48 b Fw(such)42 b(that)257 4130 y Fu(\036)315 4146 y Fq(\013;n;R)p Fs(\()p Fq(k)r(;d)p Fs(\))649 4130 y Fv(\()p Fu(C)7 b Fv(\))35 b Fw(is)g(also)f(nonzer)-5 b(o.)p Black 257 4337 a(Pr)g(o)g(of.)p Black 49 w Fx(Let)34 b Fu(C)i Fv(=)30 b Fu(T)1008 4352 y Fs(1)1070 4337 y Fv(+)23 b Ft(\001)17 b(\001)g(\001)k Fv(+)i Fu(T)1464 4352 y Fq(k)1540 4337 y Fx(b)s(e)34 b(nonzero.)47 b(W)-8 b(e)34 b(de\034ne)h(its)e Fw(gc)-5 b(d)36 b(p)-5 b(art)p Fx(,)34 b Fu(g)t(cd)p Fv(\()p Fu(C)7 b Fv(\))28 b(:=)257 4458 y Fu(g)t(cd)p Fv(\()p Fu(T)496 4473 y Fs(1)535 4458 y Fu(;)17 b(:)g(:)g(:)f(;)h(T)811 4473 y Fq(k)853 4458 y Fv(\))33 b Fx(and)f(its)g Fw(simple)i(p)-5 b(art)p Fx(,)33 b Fu(sim)p Fv(\()p Fu(C)7 b Fv(\))28 b(:=)2350 4419 y Fq(C)p 2271 4435 213 4 v 2271 4492 a(g)r(cd)p Fs(\()p Fq(C)5 b Fs(\))2494 4458 y Fx(.)404 4579 y(W)-8 b(e)26 b(no)m(w)g(de\034ne)g(some)g(subspaces)i(spanned)e(b)m(y)h(the)f (linear)e(p)s(olynomials)e(app)s(ear-)257 4699 y(ing)34 b(in)h Fu(C)7 b Fx(.)50 b(These)37 b(subspaces)g(shall)d(ha)m(v)m(e)i (the)g(prop)s(ert)m(y)f(that)g(an)m(y)h(linear)d(transfor-)257 4820 y(mation)i(whic)m(h)i(preserv)m(es)j(their)c(dimensions,)h(lea)m (v)m(es)g Fu(C)44 b Fx(nonzero.)56 b(The)38 b(subspaces)257 4940 y(are:)257 4984 y SDict begin H.S end 257 4984 a 257 4984 a SDict begin 14.5 H.A end 257 4984 a 257 4984 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 257 4984 a Black 376 5147 a Fx(1.)p Black 49 w(F)-8 b(or)45 b(ev)m(ery)j(pair)d(of)h(linear)e(forms)h Fu(`;)17 b(`)1991 5111 y Fj(0)2060 5147 y Fx(that)46 b(app)s(ear)g(in)f(the)h(circuit)f(de\034ne)501 5268 y Fu(W)593 5283 y Fq(`;`)671 5264 y Fh(0)736 5268 y Fv(:=)39 b Fu(sp)p Fv(\()p Fu(`;)17 b(`)1137 5232 y Fj(0)1160 5268 y Fv(\))p Fx(,)41 b(where)f Fu(sp)p Fv(\()p Ft(\001)p Fv(\))f Fx(refers)h(to)f(the)h(linear)d Fw(sp)-5 b(an)p Fx(,)41 b(o)m(v)m(er)f Fm(F)p Fx(,)h(of)e(the)501 5388 y(set.)p Black Black eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 257 266 a Black 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 14.5 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end 257 473 a Black 376 573 a Fx(2.)p Black 49 w(F)-8 b(or)45 b(ev)m(ery)j(nonempt)m(y)f(subset)g Fu(A)k Ft(\022)f Fv([)p Fu(k)s Fv(])p Fx(,)g(de\034ne)d Fu(C)2542 588 y Fq(A)2649 573 y Fv(:=)2803 498 y Fr(P)2908 602 y Fq(i)p Fj(2)p Fq(A)3053 573 y Fu(T)3110 588 y Fq(i)3184 573 y Fx(and)f(let)501 693 y Fu(r)545 708 y Fq(A)630 693 y Fv(:=)28 b Fu(min)p Ft(f)p Fu(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))p Fu(;)g(r)s(ank)s Fv(\()p Fu(sim)p Fv(\()p Fu(C)1851 708 y Fq(A)1908 693 y Fv(\)\))p Ft(g)p Fx(.)42 b(Let)28 b Fu(W)2365 708 y Fq(A)2451 693 y Fx(b)s(e)h(the)g(subspace)i (spanned)501 814 y(b)m(y)j Fu(r)681 829 y Fq(A)770 814 y Fx(indep)s(enden)m(t)g(linear)d(p)s(olynomials)e(that)j(app)s(ear)h (in)f Fu(sim)p Fv(\()p Fu(C)3056 829 y Fq(A)3113 814 y Fv(\))p Fx(.)257 998 y(Notice)24 b(that)h(the)g(n)m(um)m(b)s(er)g(of) f(suc)m(h)i(subspaces)g(is)e(strictly)g(less)h(than)g Fu(s)i Fv(:=)h(\()p Fu(k)3122 961 y Fs(2)3161 998 y Fu(d)3212 961 y Fs(2)3257 998 y Fv(+)6 b(2)3388 961 y Fq(k)3430 998 y Fv(\))p Fx(.)257 1118 y(The)33 b(claim)d(is)h(that)h(for)f(an)m (y)h(linear)f(transformation)e Fu(\034)39 b Fv(=)28 b Fu(\036)2533 1133 y Fq(\013;n;R)p Fs(\()p Fq(k)r(;d)p Fs(\))2899 1118 y Fx(that)j(preserv)m(es)257 1238 y(\(the)j(rank)f(of)7 b(\))33 b(all)e(these)j(subspaces,)h(also)d(satis\034es)i Fu(\034)11 b Fv(\()p Fu(C)c Fv(\))29 b Ft(6)p Fv(=)f(0)p Fx(.)45 b(W)-8 b(e)34 b(will)c(pro)m(v)m(e)35 b(this)257 1359 y(b)m(y)f(con)m(tradiction.)404 1479 y(The)39 b(\034rst)g(observ) -5 b(ation)37 b(is)h(that)g(suc)m(h)i(a)e Fu(\034)50 b Fx(cannot)38 b(map)g(t)m(w)m(o)h(linear)d(functions,)257 1599 y(that)48 b(app)s(ear)f(in)g(the)h(circuit,)j(to)c(the)h(same)g (linear)e(function)h(since)h(it)e(preserv)m(es)257 1720 y Fu(W)349 1735 y Fq(`;`)427 1716 y Fh(0)453 1720 y Fx('s.)g(Hence)34 b(the)g(simple)d(part)i(do)s(es)h(not)f(reduce)h(further)f(under)h Fu(\034)11 b Fx(,)34 b(i.e.)45 b(w)m(e)34 b(ha)m(v)m(e)257 1840 y Fu(sim)p Fv(\()p Fu(\034)11 b Fv(\()p Fu(C)620 1855 y Fq(A)678 1840 y Fv(\)\))28 b(=)j Fu(\034)11 b Fv(\()p Fu(sim)p Fv(\()p Fu(C)1252 1855 y Fq(A)1310 1840 y Fv(\)\))p Fx(.)43 b(So)31 b(w)m(e)i(can)f(assume)g(wlog)e(that)i Fu(C)38 b Fx(\(and)32 b(hence)h Fu(\034)11 b Fv(\()p Fu(C)c Fv(\))257 1961 y Fx(as)33 b(w)m(ell\))f(is)g(simple.)404 2081 y(If)24 b Fu(\034)11 b Fv(\()p Fu(C)c Fv(\))28 b(=)f(0)d Fx(then)h(the)f(rank)g(b)s(ound)g(en)m(tails)g(that)f(either)h Fu(\034)11 b Fv(\()p Fu(C)c Fv(\))24 b Fx(is)g(not)g(minimal)c(or)257 2201 y(the)35 b Fu(r)s(ank)s Fv(\()p Fu(\034)11 b Fv(\()p Fu(C)c Fv(\)\))30 b Fu(<)j(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))p Fx(.)47 b(If)34 b(the)g(latter)f(is)g(the)h(case)h(then)f Fu(r)s(ank)s Fv(\()p Fu(C)7 b Fv(\))30 b Fu(<)k(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))p Fx(,)257 2322 y(but)37 b(then)f Fu(\034)48 b Fx(will)33 b(preserv)m(e)38 b Fu(W)1417 2337 y Fs([)p Fq(k)r Fs(])1499 2322 y Fx(,)f(whic)m(h)g(means)f(that)f (the)i(circuit)e Fu(C)43 b Fx(is)35 b(itself)g(zero.)257 2442 y(This)e(con)m(tradicts)g(the)g(h)m(yp)s(othesis.)404 2562 y(The)27 b(other)g(case)h(then)f(is:)40 b Fu(\034)11 b Fv(\()p Fu(C)c Fv(\))27 b Fx(is)f(zero)h(but)g(not)f(minimal.)38 b(Let)26 b Fu(A)h Fx(b)s(e)g(a)f(minimal)257 2683 y(subset)42 b(suc)m(h)g(that)d Fu(\034)11 b Fv(\()p Fu(C)1171 2698 y Fq(A)1229 2683 y Fv(\))40 b(=)h(0)f Fx(with)f Fu(C)1812 2698 y Fq(A)1910 2683 y Ft(6)p Fv(=)h(0)p Fx(.)66 b(Hence)41 b Fu(\034)11 b Fv(\()p Fu(sim)p Fv(\()p Fu(C)2828 2698 y Fq(A)2886 2683 y Fv(\)\))40 b Fx(is)g(a)g(simple,)257 2803 y(minimal)24 b(and)29 b(zero)g(circuit,)f(therefore)h(it)e(is)h (of)g(rank)h(less)f(than)h Fu(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))p Fx(.)41 b(But)29 b(then)g Fu(\034)257 2924 y Fx(will)j(preserv)m(e)k(the)e(rank)g(of)g Fu(W)1424 2939 y Fq(A)1481 2924 y Fx(,)g(whic)m(h)g(together)g(with)f Fu(sim)p Fv(\()p Fu(C)2703 2939 y Fq(A)2761 2924 y Fv(\))c Ft(6)p Fv(=)h(0)k Fx(means)f(that)257 3044 y Fu(\034)11 b Fv(\()p Fu(sim)p Fv(\()p Fu(C)620 3059 y Fq(A)678 3044 y Fv(\)\))28 b Ft(6)p Fv(=)f(0)p Fx(.)43 b(This)33 b(is)f(again)f(a)h (con)m(tradiction.)404 3164 y(It)e(is)f(no)m(w)i(a)e(consequence)k(of)c (Lemma)1883 3164 y SDict begin H.S end 1883 3164 a Black Fx(5.2)p Black 2008 3100 a SDict begin H.R end 2008 3100 a 2008 3164 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.5.2) cvn H.B /ANN pdfmark end 2008 3164 a 29 w Fx(that)h(w)m(e)h(will) d(\034nd)i(suc)m(h)h(a)f Fu(\034)41 b Fx(if)29 b(w)m(e)i(try)257 3285 y(out)i Fu(nsR)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))835 3249 y Fs(2)906 3285 y Fx(man)m(y)33 b Fu(\013)q Fx('s.)43 b(This)33 b(\034nishes)h(the)f(pro)s(of.)p 3425 3285 4 66 v 3429 3222 59 4 v 3429 3285 V 3487 3285 4 66 v 257 3473 a Fn(Algorithm)k(&)i(Complexit)m(y)-9 b(.)102 b Fx(The)35 b(\034nal)e(blac)m(k-b)s(o)m(x)i(PIT)g(algorithm)c(is:)46 b(giv)m(en)257 3594 y(an)35 b(access)i(to)e(a)g Fv(\006\005\006\()p Fu(n;)17 b(k)s(;)g(d)p Fv(\))36 b Fx(circuit)e Fu(C)7 b Fx(,)36 b(try)f(out)g Fu(n)p Fv(2)2359 3558 y Fq(k)2402 3594 y Fu(d)2453 3558 y Fs(2)2492 3594 y Fu(R)q Fv(\()p Fu(k)s(;)17 b(d)p Fv(\))2792 3558 y Fs(2)2866 3594 y Fx(man)m(y)35 b Fu(\013)q Fx('s)g(from)257 3714 y(the)i(\034eld)g(\(or) f(its)g(extension\))h(and)g(consider)g Fu(C)2074 3678 y Fj(0)2097 3714 y Fv(\()p Fu(x)2190 3729 y Fs(1)2230 3714 y Fu(;)17 b(:)g(:)g(:)f(;)h(x)2504 3730 y Fq(R)p Fs(\()p Fq(k)r(;d)p Fs(\))2711 3714 y Fv(\))35 b(:=)h Fu(\036)2981 3730 y Fq(\013;n;R)p Fs(\()p Fq(k)r(;d)p Fs(\))3315 3714 y Fv(\()p Fu(C)7 b Fv(\))p Fx(.)257 3835 y(Ev)-5 b(aluate)33 b(eac)m(h)i(suc)m(h)g Fu(C)1182 3798 y Fj(0)1239 3835 y Fx(on)e Fv(\()p Fu(d)23 b Fv(+)g(1\))1673 3798 y Fq(R)p Fs(\()p Fq(k)r(;d)p Fs(\))1913 3835 y Fx(p)s(oin)m(ts)33 b(on)h Fm(F)2405 3798 y Fq(R)p Fs(\()p Fq(k)r(;d)p Fs(\))2612 3835 y Fx(,)g(and)f(announce)i Fu(C)41 b Fx(to)257 3955 y(b)s(e)30 b(zero)g(i\033)e(all)f(these)k(ev)-5 b(aluations)28 b(are)h(zero.)43 b(It)30 b(is)f(no)m(w)h(eviden)m(t)g(that)f(this)g(is) g(a)g(v)-5 b(alid)257 4075 y(algorithm)30 b(and)i(it)g(requires)h (\020only\021)39 b Fu(pol)r(y)t Fv(\()p Fu(n;)17 b Fv(2)2010 4039 y Fq(k)2052 4075 y Fu(;)g(d)2147 4039 y Fq(R)p Fs(\()p Fq(k)r(;d)p Fs(\))2353 4075 y Fv(\))33 b Fx(man)m(y)f Fm(F)h Fx(op)s(erations.)257 4236 y SDict begin H.S end 257 4236 a 257 4236 a SDict begin 14.5 H.A end 257 4236 a 257 4236 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.2) cvn H.B /DEST pdfmark end 257 4236 a 125 x Fi(5.2)131 b(An)44 b(almost)f(Optimal)h(Rank)g(Bound)257 4545 y Fx(The)d(b)s(est)g(kno)m (wn)g(rank)f(b)s(ound)g(for)g(minimal,)d(simple)i Fv(\006\005\006\()p Fu(n;)17 b(k)s(;)g(d)p Fv(\))40 b Fx(iden)m(tities)f(is)257 4666 y Fu(k)311 4630 y Fs(3)367 4666 y Fv(log)18 b Fu(d)26 b Fx([)p Fn(?)p Fx(].)41 b(It)27 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Fx(v)m(ertex)40 b(is)d(one)i(that)e(is)h(in)f(the)i(set)f Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))38 b Fx(\(i.e.)59 b(a)257 1301 y(form)40 b(in)g Fv(\()p Fu(U)f Ft([)28 b Fu(V)22 b Fv(\))28 b Ft(\\)g Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))p Fx(\).)69 b(Let)41 b(v)m(ertex)i Fu(v)i Fx(b)s(e)c(green,)j(so)d Fu(v)46 b Ft(2)c Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))p Fx(.)69 b(The)42 b Fu(I)3423 1316 y Fs(1)3463 1301 y Fx(-)257 1421 y(neigh)m(b)s(or)d (of)f Fu(v)k Fx(is)d(a)f(linear)f(com)m(bination)g(of)h Fu(v)43 b Fx(and)c Fu(I)2348 1436 y Fs(1)2387 1421 y Fx(.)62 b(Therefore,)42 b(the)d(neigh)m(b)s(or)257 1542 y(is)d(also)e(in)h Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))36 b Fx(and)g(is)f(colored)g(green.)54 b(This)35 b(sho)m(ws)j(that)d(the)h (n)m(um)m(b)s(er)g(of)g(green)257 1662 y(v)m(ertices)c(in)e Fu(U)41 b Fx(is)31 b(equal)f(to)g(the)h(n)m(um)m(b)s(er)g(of)g(those)g (in)f Fu(V)21 b Fx(,)31 b(at)g(the)g(end)g(of)f(eac)m(h)i(round.)404 1788 y(Let)38 b Fu(i)617 1803 y Fs(0)693 1788 y Ft(2)g Fv([)p Fu(r)s Fv(])f 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2384 y Fl(0)758 2360 y Fv(\))p Fu(;)17 b(sp)p Fv(\()p Fu(I)1016 2375 y Fq(i)1040 2384 y Fl(0)1074 2375 y Fs(+1)1169 2360 y Fv(\))p Fu(;)g(:)g(:)g(:)e(;) i(sp)p Fv(\()p Fu(I)1601 2375 y Fq(r)1639 2360 y Fv(\))35 b Fw(ar)-5 b(e)34 b(indep)-5 b(endent.)257 2571 y(Pr)g(o)g(of)43 b(of)g(Claim)947 2571 y SDict begin H.S end 947 2571 a Black Fw(5.6)p Black 1076 2507 a SDict begin H.R end 1076 2507 a 1076 2571 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.5.6) cvn H.B /ANN pdfmark end 1076 2571 a Fw(.)140 b Fx(The)42 b(forms)f Ft(f)p Fu(`)p Ft(g)p Fx(,)i Fu(I)1987 2586 y Fs(1)2027 2571 y Fu(;)17 b(:)g(:)g(:)e(;)i(I)2288 2586 y Fq(i)2312 2595 y Fl(0)2347 2586 y Fj(\000)p Fs(1)2483 2571 y Fx(are)41 b(indep)s(enden)m(t)h(b)m(y)h(the)257 2692 y(minimalit)m(y)28 b(of)k Fu(i)894 2707 y Fs(0)934 2692 y Fx(.)404 2818 y(As)d Fu(I)587 2833 y Fs(1)627 2818 y Fu(;)17 b(:)g(:)g(:)f(;)h(I)889 2833 y Fq(i)913 2842 y Fl(0)980 2818 y Fx(are)29 b(indep)s(enden)m(t)i(but)e Ft(f)p Fu(`)p Ft(g)p Fx(,)h Fu(I)2105 2833 y Fs(1)2144 2818 y Fu(;)17 b(:)g(:)g(:)f(;)h(I)2406 2833 y Fq(i)2430 2842 y Fl(0)2498 2818 y Fx(are)29 b(not,)g(w)m(e)i(deduce)f(that)257 2938 y Fu(`)e Ft(2)g Fu(sp)p Fv(\()p Fu(J)607 2953 y Fq(i)631 2962 y Fl(0)670 2938 y Fv(\))p Fx(.)42 b(Th)m(us,)32 b(the)e(subspace)i Fu(sp)p Fv(\()p Ft(f)p Fu(`)p Ft(g)16 b([)h Fu(J)2046 2953 y Fq(i)2070 2962 y Fl(0)2109 2938 y Fv(\))28 b(=)h Fu(sp)p Fv(\()p Fu(J)2467 2953 y Fq(i)2491 2962 y Fl(0)2530 2938 y Fv(\))h Fx(is)f(indep)s(enden)m(t)i(to)f(the) 257 3059 y(forms)i Fu(I)569 3074 y Fq(i)593 3083 y Fl(0)628 3074 y Fs(+1)722 3059 y Fu(;)17 b(:)g(:)g(:)f(;)h(I)984 3074 y Fq(r)1054 3059 y Fx(b)m(y)34 b(the)e(indep)s(endence)j(of)d Fu(I)2116 3074 y Fs(1)2155 3059 y Fu(;)17 b(:)g(:)g(:)f(;)h(I)2417 3074 y Fq(r)2455 3059 y Fx(.)776 b Fb(2)404 3209 y Fx(W)-8 b(e)46 b(shall)f(no)m(w)h(sho)m(w)h(that)f(for)f Fu(i)51 b Ft(62)g(f)p Fv(1)p Fu(;)17 b(i)2062 3224 y Fs(0)2101 3209 y Ft(g)p Fx(,)49 b(the)e(n)m(um)m(b)s(er)f(of)g(green)g(v)m (ertices)257 3329 y(doubles)38 b(in)e(the)i Fu(i)p Fx(th)f(round.)57 b(Let)38 b Fu(`)1622 3293 y Fj(0)1682 3329 y Fx(b)s(e)f(a)g(green)h(v)m (ertex,)i(sa)m(y)e(in)f Fu(U)10 b Fx(,)39 b(at)d(the)i(end)g(of)257 3449 y(the)32 b Fv(\()p Fu(i)18 b Ft(\000)i Fv(1\))p Fx(th)30 b(round)h(\(at)g(that)f(p)s(oin)m(t)g Fu(B)j Fv(=)27 b Ft(f)p Fu(`)p Ft(g)19 b([)g Fu(J)2226 3464 y Fq(i)p Fj(\000)p Fs(1)2344 3449 y Fx(\).)43 b(Consider)31 b(the)g Fu(I)3066 3464 y Fq(i)3094 3449 y Fx(-neigh)m(b)s(or)257 3570 y(of)j Fu(`)411 3534 y Fj(0)434 3570 y Fx(.)48 b(This)34 b(is)g(in)f Fu(V)56 b Fx(and)34 b(is)g(equal)g(to)f Fv(\()p Fu(c`)1852 3534 y Fj(0)1899 3570 y Fv(+)23 b Fu(v)t Fv(\))33 b Fx(where)j Fu(c)30 b Ft(2)h Fm(F)2634 3534 y Fj(\003)2707 3570 y Fx(and)j Fu(v)k Fx(is)33 b(a)h Fw(nonzer)-5 b(o)257 3690 y Fx(elemen)m(t)41 b(in)g Fu(sp)p Fv(\()p Fu(I)925 3705 y Fq(i)953 3690 y Fv(\))g Fx(\(since)g Fu(U;)17 b(V)62 b Fx(are)41 b(coprime\).)68 b(If)41 b(this)g(neigh)m(b)s(or)f (is)h(green,)j(then)257 3811 y Fu(v)h Fx(w)m(ould)c(b)s(e)h(a)e(linear) g(com)m(bination)f(of)h(t)m(w)m(o)i(green)g(forms,)g(implying)d Fu(v)46 b Ft(2)c Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))p Fx(.)257 3931 y(But)39 b Fu(I)500 3946 y Fq(i)567 3931 y Fx(is)g(indep)s(enden)m (t)g(to)g Fu(B)5 b Fx(,)40 b(implying)c Fu(v)42 b Ft(2)d Fu(sp)p Fv(\()p Fu(B)5 b Fv(\))26 b Ft(\\)h Fu(sp)p Fv(\()p Fu(I)2651 3946 y Fq(i)2679 3931 y Fv(\))39 b(=)f Ft(f)p Fv(0)p Ft(g)g Fx(whic)m(h)h(is)f(a)257 4051 y(con)m(tradiction.)63 b(Therefore,)42 b(the)e Fu(I)1604 4066 y Fq(i)1632 4051 y Fx(-neigh)m(b)s(or)e(of)h(an)m(y)h(green)g(v)m(ertex)h(is)e Fw(not)49 b Fx(green.)257 4172 y(On)31 b(adding)f Fu(I)779 4187 y Fq(i)838 4172 y Fx(to)g Fu(B)5 b Fx(,)31 b(the)h(n)m(um)m(b)s (er)f(of)f(green)h(v)m(ertices)h(doubles)f(\(for)f(at)g(least)h Fv(\()p Fu(r)21 b Ft(\000)e Fv(2\))257 4292 y Fx(rounds\).)404 4418 y(W)-8 b(e)25 b(started)g(o\033)f(with)g(one)h(green)g(v)m(ertex)h Fu(`)p Fx(,)h(and)d(lists)g Fu(U)10 b Fx(,)27 b Fu(V)46 b Fx(eac)m(h)25 b(of)f(size)h Fu(d)p Fx(.)41 b(Th)m(us,)257 4539 y(this)d(doubling)f(can)i(happ)s(en)g(at)f(most)g Fv(lg)18 b Fu(d)38 b Fx(times,)h(implying)c(that)k Fv(\()p Fu(r)28 b Ft(\000)f Fv(2\))37 b Ft(\024)i Fv(lg)17 b Fu(d)p Fx(.)257 4659 y(This)33 b(is)f(a)g(con)m(tradiction,)g(implying) d(that)k Fu(U;)17 b(V)53 b Fx(are)33 b(indeed)g(similar)c(lists.)p 3425 4659 4 66 v 3429 4597 59 4 v 3429 4659 V 3487 4659 4 66 v 404 4907 a(The)35 b(ab)s(o)m(v)m(e)g(lemma)d(immediately)g (implies)f(a)k(rank)f(b)s(ound)h(for)f(our)g(iden)m(tit)m(y)g Fu(C)7 b Fx(.)257 5027 y(As)27 b Fu(T)452 5042 y Fs(2)491 5027 y Fu(;)17 b(T)592 5042 y Fs(3)657 5027 y Fx(are)26 b(coprime)f(m)m(ultiplication)c(terms)k(\(b)m(y)i(simplicit)m(y)c(of)i Fu(C)7 b Fx(\))25 b(the)i(n)m(um)m(b)s(er)f(of)257 5147 y(linearly)k(indep)s(enden)m(t)k(forms)d Fu(q)36 b Fx(in)31 b Fu(T)1676 5162 y Fs(1)1748 5147 y Fx(can)h(b)s(e)g(at)g(most)g Fv(\(lg)17 b Fu(d)k Fv(+)g(2\))p Fx(.)43 b(Rep)s(eating)32 b(this)257 5268 y(argumen)m(t)38 b(wrt)g Fu(T)940 5283 y Fs(2)1018 5268 y Fx(and)g Fu(T)1270 5283 y Fs(3)1347 5268 y Fx(pro)m(v)m(es)i(that)e Fu(r)s(ank)s Fv(\()p Fu(C)7 b Fv(\))36 b(=)h Fu(O)s Fv(\(lg)17 b Fu(d)p Fv(\))p Fx(.)59 b(In)m(terestingly)-8 b(,)40 b(the)257 5388 y(com)m(binatorial) 27 b(pro)s(cedure)32 b(in)e(the)h(pro)s(of)e(of)h(Lemma)2306 5388 y SDict begin H.S end 2306 5388 a Black Fx(5.5)p Black 2431 5324 a SDict begin H.R end 2431 5324 a 2431 5388 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.5.5) cvn H.B /ANN pdfmark end 2431 5388 a 30 w Fx(also)g(suggests)i(an)e(iden)m (tit)m(y)p Black Black eop end %%Page: 18 18 TeXDict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 257 266 a SDict begin H.S end 257 266 a Black Black 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 257 266 a Black 307 x Fx(that)36 b(ac)m(hiev)m(es)i(this)e(rank)h(b)s(ound.)54 b(It)37 b(w)m(as)g(\034rst)g(constructed)h(b)m(y)f(Ka)m(y)m(al)e(&)i (Saxena)257 693 y([)p Fn(?)p Fx(]:)485 919 y Fu(C)7 b Fv(\()p Fu(x)655 934 y Fs(1)695 919 y Fu(;)17 b(:)g(:)g(:)f(;)h(x)969 934 y Fq(r)1007 919 y Fv(\))27 b(:=)1459 824 y Fr(Y)1309 1036 y Fq(b)1339 1045 y Fl(1)1373 1036 y Fq(;:::)o(;b)1502 1045 y Fk(r)r Fh(\000)p Fl(1)1615 1036 y Fj(2)p Fa(F)1703 1045 y Fl(2)1289 1108 y Fq(b)1319 1117 y Fl(1)1354 1108 y Fs(+)p Fj(\001\001\001)o Fs(+)p Fq(b)1553 1117 y Fk(r)r Fh(\000)p Fl(1)1666 1108 y Fj(\021)p Fs(1)1756 919 y Fv(\()p Fu(b)1835 934 y Fs(1)1875 919 y Fu(x)1930 934 y Fs(1)1992 919 y Fv(+)22 b Ft(\001)17 b(\001)g(\001)j Fv(+)i Fu(b)2367 934 y Fq(r)r Fj(\000)p Fs(1)2496 919 y Fu(x)2551 934 y Fq(r)r Fj(\000)p Fs(1)2680 919 y Fv(\))1295 1272 y(+)1595 1177 y Fr(Y)1445 1389 y Fq(b)1475 1398 y Fl(1)1509 1389 y Fq(;:::)o(;b)1638 1398 y Fk(r)r Fh(\000)p Fl(1)1751 1389 y Fj(2)p Fa(F)1839 1398 y Fl(2)1425 1460 y Fq(b)1455 1469 y Fl(1)1490 1460 y Fs(+)p Fj(\001\001\001)o Fs(+)p Fq(b)1689 1469 y Fk(r)r Fh(\000)p Fl(1)1802 1460 y Fj(\021)p Fs(0)1892 1272 y Fv(\()p Fu(x)1985 1287 y Fq(r)2046 1272 y Fv(+)g Fu(b)2185 1287 y Fs(1)2225 1272 y Fu(x)2280 1287 y Fs(1)2342 1272 y Fv(+)g Ft(\001)17 b(\001)g(\001)j Fv(+)i Fu(b)2717 1287 y Fq(r)r Fj(\000)p Fs(1)2846 1272 y Fu(x)2901 1287 y Fq(r)r Fj(\000)p Fs(1)3029 1272 y Fv(\))1295 1624 y(+)1595 1530 y Fr(Y)1445 1742 y Fq(b)1475 1751 y Fl(1)1509 1742 y Fq(;:::)o(;b)1638 1751 y Fk(r)r Fh(\000)p Fl(1)1751 1742 y Fj(2)p Fa(F)1839 1751 y Fl(2)1425 1813 y Fq(b)1455 1822 y Fl(1)1490 1813 y Fs(+)p Fj(\001\001\001)o Fs(+)p Fq(b)1689 1822 y Fk(r)r Fh(\000)p Fl(1)1802 1813 y Fj(\021)p Fs(1)1892 1624 y Fv(\()p Fu(x)1985 1639 y Fq(r)2046 1624 y Fv(+)g Fu(b)2185 1639 y Fs(1)2225 1624 y Fu(x)2280 1639 y Fs(1)2342 1624 y Fv(+)g Ft(\001)17 b(\001)g(\001)j Fv(+)i Fu(b)2717 1639 y Fq(r)r Fj(\000)p Fs(1)2846 1624 y Fu(x)2901 1639 y Fq(r)r Fj(\000)p Fs(1)3029 1624 y Fv(\))3295 1624 y SDict begin H.S end 3295 1624 a 3295 1624 a SDict begin 17.5 H.A end 3295 1624 a 3295 1624 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.1) cvn H.B /DEST pdfmark end 3295 1624 a Fx(\(5.1\))257 2004 y(It)33 b(can)g(b)s(e)h(seen)g(that,)f(o)m (v)m(er)h Fm(F)1397 2019 y Fs(2)1436 2004 y Fx(,)f Fu(C)40 b Fx(is)32 b(a)h(simple)e(and)i(minimal)c Fv(\006\005\006)34 b Fx(zero)f(circuit)f(of)257 2124 y(degree)e Fu(d)d Fv(=)h(2)788 2088 y Fq(r)r Fj(\000)p Fs(2)944 2124 y Fx(with)g Fu(k)j Fv(=)c(3)i Fx(m)m(ultiplication)23 b(terms)29 b(and)f Fu(r)s(ank)s Fv(\()p Fu(C)7 b Fv(\))28 b(=)f Fu(r)k Fv(=)c(lg)18 b Fu(d)c Fv(+)g(2)p Fx(.)257 2268 y Fn(In)53 b(General.)84 b Fx(The)47 b(ab)s(o)m(v)m(e)f(description)g(giv)m(es)g(a)g(fair)e (snapshot)j(of)e(the)i(general)257 2388 y(rank)32 b(b)s(ound.)44 b(F)-8 b(or)31 b(a)g(minimal,)d(simple)i Fv(\006\005\006\()p Fu(n;)17 b(k)s(;)g(d)p Fv(\))32 b Fx(iden)m(tit)m(y)f Fu(C)k Fv(=)27 b Fu(T)2979 2403 y Fs(1)3039 2388 y Fv(+)20 b Ft(\001)d(\001)g(\001)i Fv(+)h Fu(T)3425 2403 y Fq(k)3468 2388 y Fx(,)257 2509 y(w)m(e)40 b(need)f(to)f(consider)h Fw(form-ide)-5 b(als)37 b Fu(I)45 b Fv(=)37 b(\()p Fu(`)1939 2524 y Fs(1)1979 2509 y Fu(;)17 b(:)g(:)g(:)f(;)h(`)2239 2524 y Fq(k)r Fj(\000)p Fs(2)2371 2509 y Fv(\))p Fx(,)40 b(where)f(form)e Fu(`)3040 2524 y Fq(i)3107 2509 y Fx(o)s(ccurs)i(in) 257 2629 y Fu(T)314 2644 y Fq(i)343 2629 y Fx(.)58 b Fu(C)45 b Fx(mo)s(dulo)36 b Fu(I)45 b Fx(then)38 b(giv)m(es)g(us)h Fu(I)8 b Fx(-matc)m(hings)36 b(b)s(et)m(w)m(een)k Fu(T)2579 2644 y Fq(k)r Fj(\000)p Fs(1)2712 2629 y Fu(;)17 b(T)2813 2644 y Fq(k)2855 2629 y Fx(.)59 b(If)37 b(w)m(e)i(lo)s(ok)e(at)257 2749 y(suc)m(h)43 b(matc)m(hings)d(mo)s(dulo)f(sev)m(eral)i(linearly)e Fw(indep)-5 b(endent)40 b Fx(form-ideals)e Fu(I)8 b Fx('s)41 b(then)g(a)257 2870 y(generalization)28 b(of)h(Lemma)1331 2870 y SDict begin H.S end 1331 2870 a Black Fx(5.5)p Black 1456 2806 a SDict begin H.R end 1456 2806 a 1456 2870 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.5.5) cvn H.B /ANN pdfmark end 1456 2870 a 29 w Fx(sa)m(ys)i(that)f(there)g(can)g (b)s(e)g(at)f(most)g Fu(O)s Fv(\(lg)17 b Fu(d)p Fv(\))29 b Fx(indep)s(en-)257 2990 y(den)m(t)k(form-ideals.)40 b(Since)32 b(eac)m(h)g(form-ideal)c Fu(I)39 b Fx(itself)30 b(con)m(tains)i Fv(\()p Fu(k)23 b Ft(\000)d Fv(2\))31 b Fx(indep)s(enden)m(t)257 3111 y(forms,)d(this)f(suggests)i(an)e(o)m (v)m(erall)g(rank)h(b)s(ound)f(of)g Fu(O)s Fv(\()p Fu(k)19 b Fv(lg)f Fu(d)p Fv(\))p Fx(.)41 b(This)28 b(pro)s(of)e(idea)h(when)257 3231 y(formalized)36 b(gets)j(in)m(to)f(sev)m(eral)h(problems,)g(but)f (can)h(b)s(e)f(salv)-5 b(aged)38 b(to)g Fw(pr)-5 b(ove)38 b Fx(a)g(rank)257 3351 y(b)s(ound)j(of)e Fu(k)738 3315 y Fs(3)795 3351 y Fv(lg)17 b Fu(d)p Fx(.)66 b(The)42 b(highest)e(rank)g(\(minimal,)e(simple\))h(iden)m(tities)g(kno)m(wn)i (are)257 3472 y(constructed)i(using)d(Equation)1484 3472 y SDict begin H.S end 1484 3472 a Black Fx(5.1)p Black 1609 3408 a SDict begin H.R end 1609 3408 a 1609 3472 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.5.1) cvn H.B /ANN pdfmark end 1609 3472 a Fx(,)j(and)d(ha)m(v)m(e)i(rank)f Fv(\012\()p Fu(k)20 b Fv(lg)e Fu(d)p Fv(\))p Fx(.)68 b(Th)m(us,)44 b(there)d(is)f(a)257 3592 y(sligh)m(t)32 b(gap)g(in)g(our)g(understanding)h(of)f(rank.)257 3749 y SDict begin H.S end 257 3749 a 257 3749 a SDict begin 14.5 H.A end 257 3749 a 257 3749 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.3) cvn H.B /DEST pdfmark end 257 3749 a 130 x Fi(5.3)131 b(A)44 b(Rank)g(Bound)g(o)l(v)l(er)h(Reals)257 4064 y Fx(The)36 b(high)e(rank)h(iden)m(tities)f(that)g(w)m(e)i(sa)m(w) f(in)f(the)h(last)f(section)h(are)g(o)m(v)m(er)h(\034elds)f(with)257 4184 y Fw(nonzer)-5 b(o)37 b Fx(c)m(haracteristic.)58 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