Content-Type: multipart/mixed; boundary="-------------0712040922555" This is a multi-part message in MIME format. ---------------0712040922555 Content-Type: text/plain; name="07-302.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-302.keywords" Cauchy problem, functional differential equation, q-difference equation, entire function with q-exponential growth, transseries expansion, singularity analysis ---------------0712040922555 Content-Type: application/postscript; name="qdiffsingana.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="qdiffsingana.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: qdiffsingana.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMMI12 CMSY10 CMR12 CMBX10 CMR10 CMMI10 CMBX12 %%+ CMMI8 CMSY8 CMR8 CMEX10 CMR6 CMMI6 MSBM10 CMTI10 MSBM7 LASY10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o qdiffsingana.ps qdiffsingana.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2007.12.04:1602 %%BeginProcSet: tex.pro 0 0 %! /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/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]{Ci}imagemask restore}B/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 %%BeginProcSet: texps.pro 0 0 %! 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%%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. 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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 113 /q put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 51 /three put dup 52 /four put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 79 /O put dup 83 /S put dup 84 /T put dup 87 /W put dup 94 /circumflex 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 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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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 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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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 12 /beta put dup 59 /comma put dup 83 /S put dup 104 /h put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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2AEC678647D42E788673C252C0F94C817B353C4EB1B57B483F6E0FB2F63A9C1B 972F1077FD3815AB88D7FFDA13B4BC5BA299523DE1ED18C49039CDED7DE8C237 908DC78B4E121458A5494FDFF516439F29893092F74982687F971EC529A71112 B2B9595ED4A82C5359DA04476EB09CBEBF79AF670DF92531EAE544C50F448711 0CD917272656655C7D881ADBCBC12AA93AB9FD5545501CED500834A2D61B9DBC 8F218CF8BE436F9425FF913BA99308436AE3AD8F5249D2700B342B60D112B134 2B30B9C65BCA2AB20298D64C2C5BC1B7F97F65FEFFB5E9D3454EA06B83817253 AEE6146B6800E67C0A092A2CDC2422FA101460B5037FE312CCC38E5ABCEF36CC EA810D6D56A02F033924A6669AF648C5D950EBFA6F7F420CFA67C06F33F01F7E 13F8883744554E62CF04056DBBE20EAA3B0D605532343549155F3921E1018692 D8F30E50CB2B08C3E281EE0B80FF1E1B1E98E4C6033E6DE4F33A22876F0ECC98 CE55118AC82F7B1480FFAACEA54C8222E45615C5DD5E84C28D3606D066997375 9B61FEF6FBB18EFC0EF438F5F4C8D38D840BB3509516835731396FF526F9CD13 7281B43D1C0F84B0927F1736FE0264137BD5825397BBA12E70242FAD3B9C99AC 9C04BC4A2E8892449B2E0FB7039749E1011947E07BF2C6783A663147A13F0910 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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 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 12 /vextendsingle put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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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 22 /macron put 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 61 /equal put dup 97 /a put dup 100 /d put dup 101 /e put dup 103 /g put dup 108 /l put dup 118 /v put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 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 3 /asteriskmath put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 80 /P put dup 83 /S put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 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 12 /beta put dup 14 /delta put dup 20 /kappa put dup 22 /mu put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 66 /B put dup 72 /H put dup 73 /I put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 88 /X put dup 100 /d put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n 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 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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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 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 22 /mu put dup 23 /nu put dup 28 /tau put dup 30 /phi put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 2 /Theta put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D 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 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex 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 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 /ff put dup 45 /hyphen put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 73 /I put dup 82 /R put dup 84 /T 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 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 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 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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. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 54 /six put dup 55 /seven put dup 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V 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 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % 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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y Fm(j)2722 2420 y Fn(j)2782 2357 y Fo(>)g Fp(0)p Fo(;)75 2600 y Fp(for)30 b(1)c Fn(\024)f Fo(k)j Fn(\024)d Fp(deg)r(\()p Fo(Q)p Fp(\).)41 b(In)30 b(particular,)h Fo(u)p Fp(\(1)p Fo(=q)1678 2567 y Fm(n)1726 2600 y Fo(t)p Fp(\))f(is)h(w)m(ell)g(de\014ned)e(for)h(all)i Fo(t)25 b Fn(2)2843 2578 y Fp(\026)2822 2600 y Fn(S)7 b Fp(.)216 2757 y(There)24 b(exist)h Fo(M)36 b(>)25 b Fp(0)g(and)e Fo(A)j(>)f Fp(0)g(suc)m(h)f(that)h(for)f(all)h Fo(z)30 b Fn(2)25 b Fg(C)p Fp(,)g Fn(j)p Fo(z)t Fn(j)h(\025)f Fo(A)g Fp(implies)g Fn(j)p Fo(P)13 b Fp(\()p Fo(z)t Fp(\))p Fn(j)26 b(\024)f Fo(M)10 b Fn(j)p Fo(z)t Fn(j)3486 2724 y Fk(deg)s(\()p Fm(P)g Fk(\))3708 2757 y Fp(.)38 b(In)75 2869 y(particular,)23 b(there)e(exists)g Fo(h)1028 2883 y Fk(0)p Fm(;n)1156 2869 y Fo(<)k Fp(0)c(dep)s(ending)e(on)h Fo(n)g Fp(suc)m(h)h(that)g(for)f(all)i Fo(h)j Fn(\024)g Fo(h)2790 2883 y Fk(0)p Fm(;n)2893 2869 y Fp(,)d(for)f(all)g Fo(x)3238 2883 y Fm(j)3300 2869 y Fn(2)3407 2846 y Fp(\026)3386 2869 y Fo(B)t Fp(\(1)p Fo(=n)3639 2883 y Fm(j)3677 2869 y Fo(;)15 b(\016)3757 2883 y Fm(j)3794 2869 y Fp(\),)75 2982 y(w)m(e)31 b(ha)m(v)m(e)g(1)p Fo(=)p Fp(\()p Fn(j)p Fo(q)s Fn(j)637 2949 y Fm(n)p Fk(+)p Fm(h)782 2982 y Fn(j)p Fo(x)859 2996 y Fm(j)895 2982 y Fn(j)p Fp(\))26 b Fn(\025)f Fo(A)31 b Fp(whic)m(h)f(implies)75 3209 y(\(5\))1066 b Fn(j)p Fo(P)13 b Fp(\()1512 3147 y(1)p 1397 3188 275 4 v 1397 3271 a Fo(q)1441 3245 y Fm(n)p Fk(+)p Fm(h)1583 3271 y Fo(x)1635 3285 y Fm(j)1682 3209 y Fp(\))p Fn(j)26 b(\024)f Fo(M)2283 3147 y Fp(1)p 1972 3188 667 4 v 1972 3275 a(\()p Fn(j)p Fo(q)s Fn(j)2101 3248 y Fm(n)p Fk(+)p Fm(h)2244 3275 y Fn(j)p Fo(x)2321 3289 y Fm(j)2358 3275 y Fn(j)p Fp(\))2418 3248 y Fk(deg)r(\()p Fm(P)10 b Fk(\))2649 3209 y Fo(:)75 3449 y Fp(F)-8 b(rom)31 b(\(4\))g(and)f(\(5\),)i(w)m(e)f (deduce)1049 3696 y Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1330 3659 y Fm(n)p Fk(+)p Fm(h)1474 3696 y Fo(x)1526 3710 y Fm(j)1562 3696 y Fp(\)\))p Fn(j)c(\024)1925 3635 y Fo(M)p 1789 3675 369 4 v 1789 3762 a Fn(j)p Fo(a)p Fn(j)p Fo(\021)1935 3736 y Fk(deg)s(\()p Fm(Q)p Fk(\))2178 3634 y Fp(\()p Fn(j)p Fo(q)s Fn(j)2307 3601 y Fm(n)p Fk(+)p Fm(h)2450 3634 y Fn(j)p Fo(x)2527 3648 y Fm(j)2564 3634 y Fn(j)p Fp(\))2624 3601 y Fk(deg)r(\()p Fm(Q)p Fk(\))p 2178 3675 668 4 v 2178 3762 a Fp(\()p Fn(j)p Fo(q)s Fn(j)2307 3736 y Fm(n)p Fk(+)p Fm(h)2450 3762 y Fn(j)p Fo(x)2527 3776 y Fm(j)2564 3762 y Fn(j)p Fp(\))2624 3736 y Fk(deg)s(\()p Fm(P)10 b Fk(\))2856 3696 y Fo(:)75 3951 y Fp(F)-8 b(rom)45 b Fc(\(H1\))p Fp(,)j(the)d(sequence)g Fn(j)p Fo(q)s Fn(j)1273 3918 y Fm(h)p Fk(\(deg)r(\()p Fm(Q)p Fk(\))p Fl(\000)p Fk(deg)r(\()p Fm(P)10 b Fk(\)\))1906 3951 y Fp(is)45 b(b)s(ounded)d(as)j Fo(h)50 b Fn(!)f(\0001)p Fp(.)83 b(So)45 b(that)g(there)g(exists)75 4064 y Fo(M)163 4078 y Fk(0)p Fm(;n)295 4064 y Fo(>)30 b Fp(0)k(dep)s(ending)e(on)h Fo(n)p Fp(,)h(there)g(exists)g Fo(h)1700 4078 y Fk(0)p Fm(;n)1832 4064 y Fo(<)c Fp(0)k(dep)s(ending)e(on)h Fo(n)g Fp(suc)m(h)g(that)h(for)f(all)i Fo(h)30 b Fn(\024)g Fo(h)3584 4078 y Fk(0)p Fm(;n)3686 4064 y Fp(,)35 b(for)75 4177 y(all)c Fo(x)253 4191 y Fm(j)315 4177 y Fn(2)422 4154 y Fp(\026)401 4177 y Fo(B)t Fp(\(1)p Fo(=n)654 4191 y Fm(j)692 4177 y Fo(;)15 b(\016)772 4191 y Fm(j)809 4177 y Fp(\),)75 4361 y(\(6\))1302 b Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1773 4324 y Fm(n)p Fk(+)p Fm(h)1917 4361 y Fo(x)1969 4375 y Fm(j)2005 4361 y Fp(\)\))p Fn(j)27 b(\024)e Fo(M)2311 4375 y Fk(0)p Fm(;n)2413 4361 y Fo(:)75 4545 y Fp(There)34 b(exist)i Fo(M)42 b(>)32 b Fp(0)j(and)f Fo(\016)j(>)32 b Fp(0)j(suc)m(h)f(that)h(for)g(all)g Fo(z)i Fn(2)32 b Fg(C)p Fp(,)j Fn(j)p Fo(z)t Fn(j)f Fo(<)e(\016)38 b Fp(implies)d Fn(j)p Fo(P)13 b Fp(\()p Fo(z)t Fp(\))p Fn(j)34 b(\024)e Fo(M)10 b Fn(j)p Fo(z)t Fn(j)3490 4512 y Fk(v)l(al)q(\()p Fm(P)g Fk(\))3693 4545 y Fp(.)53 b(In)75 4658 y(particular,)23 b(there)e(exists)g Fo(h)1028 4672 y Fk(1)p Fm(;n)1156 4658 y Fo(>)k Fp(0)c(dep)s(ending)e(on)h Fo(n)g Fp(suc)m(h)h(that)g(for)f(all)i Fo(h)j Fn(\025)g Fo(h)2790 4672 y Fk(1)p Fm(;n)2893 4658 y Fp(,)d(for)f(all)g Fo(x)3238 4672 y Fm(j)3300 4658 y Fn(2)3407 4635 y Fp(\026)3386 4658 y Fo(B)t Fp(\(1)p Fo(=n)3639 4672 y Fm(j)3677 4658 y Fo(;)15 b(\016)3757 4672 y Fm(j)3794 4658 y Fp(\),)75 4771 y(w)m(e)31 b(ha)m(v)m(e)g(1)p Fo(=)p Fp(\()p Fn(j)p Fo(q)s Fn(j)637 4738 y Fm(n)p Fk(+)p Fm(h)782 4771 y Fn(j)p Fo(x)859 4785 y Fm(j)895 4771 y Fn(j)p Fp(\))26 b Fn(\024)f Fo(\016)34 b Fp(whic)m(h)c(implies)75 4997 y(\(7\))1075 b Fn(j)p Fo(P)13 b Fp(\()1521 4936 y(1)p 1406 4976 275 4 v 1406 5060 a Fo(q)1450 5033 y Fm(n)p Fk(+)p Fm(h)1593 5060 y Fo(x)1645 5074 y Fm(j)1691 4997 y Fp(\))p Fn(j)26 b(\024)f Fo(M)2283 4936 y Fp(1)p 1981 4976 649 4 v 1981 5064 a(\()p Fn(j)p Fo(q)s Fn(j)2110 5037 y Fm(n)p Fk(+)p Fm(h)2253 5064 y Fn(j)p Fo(x)2330 5078 y Fm(j)2367 5064 y Fn(j)p Fp(\))2427 5037 y Fk(v)l(al)q(\()p Fm(P)10 b Fk(\))2639 4997 y Fo(:)75 5249 y Fp(F)-8 b(rom)33 b Fc(\(H1\))f Fp(w)m(e)h(ha)m(v)m(e)h Fo(d)959 5264 y Fm(k)1031 5249 y Fn(6)p Fp(=)29 b(0,)34 b(1)29 b Fn(\024)g Fo(k)j Fn(\024)d Fp(deg)q(\()p Fo(Q)p Fp(\))34 b(and)e(from)g(the)h (fact)g(that)g(lim)2963 5264 y Fm(h)p Fl(!)p Fk(+)p Fl(1)3219 5249 y Fp(1)p Fo(=)p Fp(\()p Fo(q)3388 5216 y Fm(n)p Fk(+)p Fm(h)3531 5249 y Fo(x)3583 5263 y Fm(j)3620 5249 y Fp(\))c(=)g(0,)75 5362 y(w)m(e)k(deduce)f(that)h(there)f(exist)i Fo(\021)1212 5376 y Fk(1)1280 5362 y Fo(>)28 b Fp(0,)33 b Fo(h)1534 5376 y Fk(1)p Fm(;n)1665 5362 y Fo(>)c Fp(0)j(suc)m(h)g (that)h(for)f(all)i Fo(h)29 b Fn(\025)f Fo(h)2750 5376 y Fk(1)p Fm(;n)2852 5362 y Fp(,)33 b(for)f(all)i Fo(x)3232 5376 y Fm(j)3297 5362 y Fn(2)3407 5339 y Fp(\026)3386 5362 y Fo(B)t Fp(\(1)p Fo(=n)3639 5376 y Fm(j)3677 5362 y Fo(;)15 b(\016)3757 5376 y Fm(j)3794 5362 y Fp(\),)75 5475 y(1)26 b Fn(\024)f Fo(k)j Fn(\024)d Fp(deg)r(\()p Fo(Q)p Fp(\),)75 5697 y(\(8\))1399 b Fn(j)p Fo(d)1661 5712 y Fm(k)1724 5697 y Fn(\000)1940 5635 y Fp(1)p 1825 5676 275 4 v 1825 5759 a Fo(q)1869 5733 y Fm(n)p Fk(+)p Fm(h)2011 5759 y Fo(x)2063 5773 y Fm(j)2110 5697 y Fn(j)25 b(\025)g Fo(\021)2301 5711 y Fk(1)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fp(4)75 399 y(F)-8 b(rom)31 b(\(8\))g(and)f(\(7\),)i(w)m(e)f(deduce)1028 629 y Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1309 592 y Fm(n)p Fk(+)p Fm(h)1453 629 y Fo(x)1505 643 y Fm(j)1541 629 y Fp(\)\))p Fn(j)c(\024)2281 568 y Fo(M)p 1768 608 1124 4 v 1768 696 a Fn(j)p Fo(a)p Fn(j)p Fp(\()p Fo(\021)1946 710 y Fk(1)1987 696 y Fp(\))2022 669 y Fk(deg)r(\()p Fm(Q)p Fk(\))2244 696 y Fp(\()p Fn(j)p Fo(q)s Fn(j)2373 669 y Fm(n)p Fk(+)p Fm(h)2516 696 y Fn(j)p Fo(x)2593 710 y Fm(j)2630 696 y Fn(j)p Fp(\))2690 669 y Fk(v)l(al\()p Fm(P)10 b Fk(\))75 889 y Fp(F)-8 b(rom)35 b Fc(\(H1\))p Fp(,)g(w)m(e)f(ha)m(v)m(e)i(v)-5 b(al\()p Fo(P)13 b Fp(\))33 b Fn(\025)e Fp(1,)36 b(so)e(that)h(1)p Fo(=)p Fn(j)p Fo(q)s Fn(j)1940 857 y Fm(h)p Fk(v)l(al)q(\()p Fm(P)10 b Fk(\))2218 889 y Fp(tends)34 b(to)h(0)g(as)f Fo(h)e Fn(!)g Fp(+)p Fn(1)p Fp(.)52 b(W)-8 b(e)35 b(deduce)f(that)75 1002 y(there)i(exists)g Fo(M)657 1016 y Fk(1)p Fm(;n)793 1002 y Fo(>)e Fp(0)i(dep)s(ending)e(on)i Fo(n)p Fp(,)g(there)g(exists)h Fo(h)2216 1016 y Fk(1)p Fm(;n)2352 1002 y Fo(>)d Fp(0)i(dep)s(ending)e (on)h Fo(n)h Fp(suc)m(h)f(that)h(for)g(all)75 1115 y Fo(h)26 b Fn(\025)f Fo(h)301 1129 y Fk(1)p Fm(;n)403 1115 y Fp(,)30 b(for)h(all)g Fo(x)776 1129 y Fm(j)838 1115 y Fn(2)945 1092 y Fp(\026)923 1115 y Fo(B)5 b Fp(\(1)p Fo(=n)1177 1129 y Fm(j)1214 1115 y Fo(;)15 b(\016)1294 1129 y Fm(j)1332 1115 y Fp(\),)75 1304 y(\(9\))1302 b Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1773 1267 y Fm(n)p Fk(+)p Fm(h)1917 1304 y Fo(x)1969 1318 y Fm(j)2005 1304 y Fp(\)\))p Fn(j)27 b(\024)e Fo(M)2311 1318 y Fk(1)p Fm(;n)2413 1304 y Fo(:)75 1494 y Fp(F)-8 b(rom)31 b(\(4\),)h Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)742 1461 y Fm(n)p Fk(+)p Fm(h)886 1494 y Fo(x)938 1508 y Fm(j)974 1494 y Fp(\)\))f(is)g(w)m(ell)g(de\014ned)f(for)g(all)h Fo(h)1982 1508 y Fk(0)p Fm(;n)2110 1494 y Fn(\024)25 b Fo(h)h Fn(\024)f Fo(h)2432 1508 y Fk(1)p Fm(;n)2534 1494 y Fp(,)31 b(all)g Fo(x)2768 1508 y Fm(j)2830 1494 y Fn(2)2937 1471 y Fp(\026)2916 1494 y Fo(B)t Fp(\(1)p Fo(=n)3169 1508 y Fm(j)3207 1494 y Fo(;)15 b(\016)3287 1508 y Fm(j)3324 1494 y Fp(\).)41 b(W)-8 b(e)32 b(deduce)75 1607 y(that)27 b(there)f(exists)g Fo(M)830 1621 y Fk(2)p Fm(;n)958 1607 y Fo(>)e Fp(0)j(dep)s(ending)d (on)i Fo(n)f Fp(suc)m(h)h(that)h(for)e(all)i Fo(h)2461 1621 y Fk(0)p Fm(;n)2589 1607 y Fn(\024)e Fo(h)g Fn(\024)g Fo(h)2910 1621 y Fk(1)p Fm(;n)3012 1607 y Fp(,)i(all)g Fo(x)3238 1621 y Fm(j)3300 1607 y Fn(2)3407 1584 y Fp(\026)3386 1607 y Fo(B)t Fp(\(1)p Fo(=n)3639 1621 y Fm(j)3677 1607 y Fo(;)15 b(\016)3757 1621 y Fm(j)3794 1607 y Fp(\),)75 1796 y(\(10\))1257 b Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1773 1758 y Fm(n)p Fk(+)p Fm(h)1917 1796 y Fo(x)1969 1810 y Fm(j)2005 1796 y Fp(\)\))p Fn(j)27 b(\024)e Fo(M)2311 1810 y Fk(2)p Fm(;n)2413 1796 y Fo(:)75 1985 y Fp(Finally)-8 b(,)30 b(from)d(\(6\),)i(\(9\),)g(\(10\),)h(w)m(e)e(get)h Fo(M)1533 1999 y Fm(n)1605 1985 y Fp(=)c(sup)o(\()p Fo(M)1961 1999 y Fk(0)p Fm(;n)2063 1985 y Fo(;)15 b(M)2191 1999 y Fk(1)p Fm(;n)2294 1985 y Fo(;)g(M)2422 1999 y Fk(2)p Fm(;n)2524 1985 y Fp(\))28 b(dep)s(ending)e(on)h Fo(n)g Fp(suc)m(h)g(that)h(for)g(all)75 2098 y Fo(h)e Fn(2)e Fg(Z)p Fp(,)31 b(for)f(all)h Fo(x)672 2112 y Fm(j)734 2098 y Fn(2)841 2075 y Fp(\026)820 2098 y Fo(B)t Fp(\(1)p Fo(=n)1073 2112 y Fm(j)1111 2098 y Fo(;)15 b(\016)1191 2112 y Fm(j)1228 2098 y Fp(\),)31 b(1)25 b Fn(\024)g Fo(j)31 b Fn(\024)25 b Fo(s)p Fp(,)1492 2287 y Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1773 2249 y Fm(n)p Fk(+)p Fm(h)1917 2287 y Fo(x)1969 2301 y Fm(j)2005 2287 y Fp(\)\))p Fn(j)i(\024)e Fo(M)2311 2301 y Fk(3)p Fm(;n)2413 2287 y Fo(:)75 2476 y Fp(The)30 b(part)g(a\))h(of)g(Lemma)f(1)h(follo)m (ws.)216 2634 y(b\))44 b(Let)g Fo(h)49 b Fn(2)e Fg(Z)p Fp(.)82 b(F)-8 b(rom)44 b Fc(\(Q1\))f Fp(w)m(e)i(ha)m(v)m(e)g Fo(d)1825 2649 y Fm(k)1916 2634 y Fn(6)p Fp(=)j(0,)g(1)g Fn(\024)g Fo(k)j Fn(\024)d Fp(deg)q(\()p Fo(Q)p Fp(\))d(and)e(from)h (the)g(fact)h(that)75 2747 y(lim)201 2761 y Fm(n)p Fl(!)p Fk(+)p Fl(1)459 2747 y Fp(1)p Fo(=)p Fp(\()p Fo(q)628 2714 y Fm(n)p Fk(+)p Fm(h)772 2747 y Fo(x)824 2761 y Fm(j)861 2747 y Fp(\))27 b(=)g(0,)32 b(there)g(exists)g Fo(\021)1654 2761 y Fk(1)1720 2747 y Fo(>)27 b Fp(0,)32 b(there)g(exists)g Fo(n)2461 2762 y Fk(0)p Fm(;h)2587 2747 y Fn(\025)27 b Fp(0)32 b(\(dep)s(ending)e(on)i Fo(h)p Fp(\))g(suc)m(h)f(that)75 2860 y(for)f(all)h Fo(n)25 b Fn(\025)g Fo(n)571 2875 y Fk(0)p Fm(;h)670 2860 y Fp(,)31 b(for)f(all)h Fo(x)1043 2874 y Fm(j)1105 2860 y Fn(2)1213 2837 y Fp(\026)1191 2860 y Fo(B)5 b Fp(\(1)p Fo(=n)1445 2874 y Fm(j)1482 2860 y Fo(;)15 b(\016)1562 2874 y Fm(j)1599 2860 y Fp(\),)31 b(1)26 b Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(s)p Fp(,)75 3092 y(\(11\))1354 b Fn(j)p Fo(d)1661 3107 y Fm(k)1724 3092 y Fn(\000)1940 3030 y Fp(1)p 1825 3071 275 4 v 1825 3154 a Fo(q)1869 3128 y Fm(n)p Fk(+)p Fm(h)2011 3154 y Fo(x)2063 3168 y Fm(j)2110 3092 y Fn(j)25 b(\025)g Fo(\021)2301 3106 y Fk(1)75 3334 y Fp(On)40 b(the)h(other)g(hand,)i(there)e(exist)h Fo(M)53 b(>)42 b Fp(0)g(and)e Fo(\016)47 b(>)42 b Fp(0)g(suc)m(h)e(that)i(for)f(all)g Fo(z)47 b Fn(2)c Fg(C)p Fp(,)g Fn(j)p Fo(z)t Fn(j)h Fo(<)f(\016)h Fp(implies)75 3446 y Fn(j)p Fo(P)13 b Fp(\()p Fo(z)t Fp(\))p Fn(j)43 b(\024)f Fo(M)10 b Fn(j)p Fo(z)t Fn(j)662 3414 y Fk(v)l(al)q(\()p Fm(P)g Fk(\))865 3446 y Fp(.)71 b(In)39 b(particular,)44 b(there)d(exists)g Fo(n)2101 3461 y Fk(0)p Fm(;h)2242 3446 y Fn(\025)h Fp(0)f(\(dep)s(ending)e(on)h Fo(h)p Fp(\))h(suc)m(h)f(that)h(for)g(all)75 3559 y Fo(n)25 b Fn(\025)g Fo(n)306 3574 y Fk(0)p Fm(;h)405 3559 y Fp(,)31 b(for)f(all)h Fo(x)778 3573 y Fm(j)840 3559 y Fn(2)947 3536 y Fp(\026)926 3559 y Fo(B)t Fp(\(1)p Fo(=n)1179 3573 y Fm(j)1216 3559 y Fo(;)15 b(\016)1296 3573 y Fm(j)1334 3559 y Fp(\))30 b(implies)h(1)p Fo(=)p Fp(\()p Fn(j)p Fo(q)s Fn(j)1927 3526 y Fm(n)p Fk(+)p Fm(h)2071 3559 y Fn(j)p Fo(x)2148 3573 y Fm(j)2185 3559 y Fn(j)p Fp(\))26 b Fn(\024)f Fo(\016)s Fp(,)31 b(whic)m(h)f(implies)75 3792 y(\(12\))597 b Fn(j)p Fo(P)13 b Fp(\()1089 3730 y(1)p 973 3771 V 973 3854 a Fo(q)1017 3828 y Fm(n)p Fk(+)p Fm(h)1160 3854 y Fo(x)1212 3868 y Fm(j)1259 3792 y Fp(\))p Fn(j)26 b(\024)f Fo(M)1850 3730 y Fp(1)p 1549 3771 649 4 v 1549 3858 a(\()p Fn(j)p Fo(q)s Fn(j)1678 3832 y Fm(n)p Fk(+)p Fm(h)1821 3858 y Fn(j)p Fo(x)1898 3872 y Fm(j)1935 3858 y Fn(j)p Fp(\))1995 3832 y Fk(v)l(al\()p Fm(P)10 b Fk(\))2232 3792 y Fn(\024)25 b Fo(M)2726 3730 y Fp(1)p 2436 3771 626 4 v 2436 3858 a(\()p Fn(j)p Fo(q)s Fn(j)2565 3832 y Fm(n)p Fk(+)p Fm(h)2708 3858 y Fo(m)2788 3872 y Fm(j)2825 3858 y Fp(\))2860 3832 y Fk(v)l(al\()p Fm(P)10 b Fk(\))3072 3792 y Fo(:)75 4044 y Fp(where)30 b Fo(m)418 4058 y Fm(j)479 4044 y Fo(>)25 b Fp(0)31 b(dep)s(ends)e(on)1151 4021 y(\026)1129 4044 y Fo(B)t Fp(\(1)p Fo(=n)1382 4058 y Fm(j)1420 4044 y Fo(;)15 b(\016)1500 4058 y Fm(j)1537 4044 y Fp(\).)216 4157 y(F)-8 b(rom)31 b(\(11\))h(and)e(\(12\),)i(w)m (e)f(deduce)f(that)1039 4387 y Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1320 4350 y Fm(n)p Fk(+)p Fm(h)1464 4387 y Fo(x)1516 4401 y Fm(j)1553 4387 y Fp(\)\))p Fn(j)c(\024)2281 4326 y Fo(M)p 1780 4367 1101 4 v 1780 4454 a Fn(j)p Fo(a)p Fn(j)p Fp(\()p Fo(\021)1958 4468 y Fk(1)1998 4454 y Fp(\))2033 4427 y Fk(deg)r(\()p Fm(Q)p Fk(\))2255 4454 y Fp(\()p Fn(j)p Fo(q)s Fn(j)2384 4427 y Fm(n)p Fk(+)p Fm(h)2527 4454 y Fo(m)2607 4468 y Fm(j)2644 4454 y Fp(\))2679 4427 y Fk(v)l(al\()p Fm(P)10 b Fk(\))75 4639 y Fp(for)30 b(all)h Fo(n)25 b Fn(\025)g Fo(n)571 4654 y Fk(0)p Fm(;h)670 4639 y Fp(,)31 b(for)f(all)h Fo(x)1043 4653 y Fm(j)1105 4639 y Fn(2)1213 4616 y Fp(\026)1191 4639 y Fo(B)5 b Fp(\(1)p Fo(=n)1445 4653 y Fm(j)1482 4639 y Fo(;)15 b(\016)1562 4653 y Fm(j)1599 4639 y Fp(\),)31 b(1)26 b Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(s)p Fp(.)40 b(The)30 b(part)g(b\))g(of)h (Lemma)f(1)h(follo)m(ws.)331 b Fa(2)75 4882 y Fc(Prop)s(osition)36 b(1)46 b Ff(a\))33 b(The)f(in\014nite)h(pr)-5 b(o)g(duct)1674 4813 y Fj(Q)1760 4840 y Fk(+)p Fl(1)1760 4908 y Fm(n)p Fk(=0)1897 4882 y Fp(\(1)21 b(+)e Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)2344 4849 y Fm(n)2393 4882 y Fo(t)p Fp(\)\)\))33 b Ff(c)-5 b(onver)g(ges)33 b(uniformly)g(on)g(the)g(c)-5 b(om-)75 4995 y(p)g(act)34 b(sets)f Fn([)508 4962 y Fm(s)508 5020 y(j)t Fk(=1)634 4995 y Fo(q)678 4962 y Fm(h)744 4972 y Fp(\026)723 4995 y Fo(B)t Fp(\(1)p Fo(=n)976 5009 y Fm(j)1014 4995 y Fo(;)15 b(\016)1094 5009 y Fm(j)1131 4995 y Fp(\))33 b Ff(for)g(any)g Fo(h)26 b Fn(2)f Fg(Z)p Ff(.)75 5120 y(b\))33 b(We)f(denote)h(by)g Fp(\002)25 b(=)f Fn([)1013 5087 y Fm(s)1013 5145 y(j)t Fk(=1)1170 5084 y(1)p 1150 5099 76 4 v 1150 5151 a Fm(n)1193 5161 y Fh(j)1235 5120 y Fo(q)1279 5087 y Fl(\000)p Fb(N)1386 5120 y Ff(.)42 b(Then,)32 b(the)h(function)1360 5376 y Fo(e)1402 5390 y Fm(q)1441 5376 y Fp(\()p Fo(t)p Fp(\))26 b(=)2082 5314 y(1)p 1676 5355 859 4 v 1676 5380 a Fj(Q)1761 5406 y Fk(+)p Fl(1)1761 5475 y Fm(n)p Fk(=0)1899 5448 y Fp(\(1)21 b(+)e Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)2346 5421 y Fm(n)2395 5448 y Fo(t)p Fp(\)\)\))2544 5376 y Fo(;)75 5627 y Ff(is)26 b(a)h(solution)h(of)e(\(3\))h(which)g(is)g (holomorphic)i(on)e(the)g(op)-5 b(en)27 b(set)g Fn(S)13 b(n)7 b Fp(\002)26 b Ff(and)h(has)g(mer)-5 b(omorphic)30 b(singularities)75 5740 y(on)j Fp(\002)p Ff(.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fp(5)75 399 y Fc(Pro)s(of)31 b Fp(a\))f(Let)g Fo(h)c Fn(2)f Fg(Z)p Fp(.)40 b(F)-8 b(rom)30 b(Lemma)g(1\))g(b\),)g(there)g(exist)g Fo(M)2268 414 y Fm(h)2338 399 y Fo(>)25 b Fp(0)30 b(and)g Fo(n)2741 414 y Fk(0)p Fm(;h)2865 399 y Fo(>)25 b Fp(0)30 b(dep)s(ending)e(on)i Fo(h)g Fp(suc)m(h)75 511 y(that)1379 640 y Fn(j)p Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1660 602 y Fm(n)1709 640 y Fo(t)p Fp(\)\))p Fn(j)c(\024)f Fo(M)2047 655 y Fm(h)2092 640 y Fp(\()2185 578 y(1)p 2137 619 142 4 v 2137 702 a Fn(j)p Fo(q)s Fn(j)2231 676 y Fm(n)2288 640 y Fp(\))2323 602 y Fk(v)l(al)q(\()p Fm(P)10 b Fk(\))2525 640 y Fo(;)75 866 y Fp(for)26 b(all)h Fo(n)e Fn(\025)f Fo(n)562 881 y Fk(0)p Fm(;h)662 866 y Fp(,)j(for)f(all)h Fo(t)e Fn(2)f([)1175 833 y Fm(s)1175 892 y(j)t Fk(=1)1302 866 y Fo(q)1346 833 y Fm(h)1412 843 y Fp(\026)1390 866 y Fo(B)5 b Fp(\(1)p Fo(=n)1644 880 y Fm(j)1681 866 y Fo(;)15 b(\016)1761 880 y Fm(j)1799 866 y Fp(\).)39 b(W)-8 b(e)27 b(deduce)f(that)h(there)f (exist)h Fo(M)3069 881 y Fk(1)p Fm(;h)3194 866 y Fo(>)e Fp(0)h(and)g Fo(n)3589 881 y Fk(1)p Fm(;h)3713 866 y Fo(>)f Fp(0)75 979 y(suc)m(h)30 b(that)1247 1108 y Fn(j)p Fp(Log)r(\(1)21 b(+)f Fo(u)p Fp(\()1749 1046 y(1)p 1710 1087 124 4 v 1710 1170 a Fo(q)1754 1144 y Fm(n)1801 1170 y Fo(t)1844 1108 y Fp(\)\))p Fn(j)26 b(\024)f Fo(M)2149 1123 y Fk(1)p Fm(;h)2249 1108 y Fp(\()2342 1046 y(1)p 2294 1087 142 4 v 2294 1170 a Fn(j)p Fo(q)s Fn(j)2388 1144 y Fm(n)2446 1108 y Fp(\))2481 1070 y Fk(v)l(al\()p Fm(P)10 b Fk(\))75 1334 y Fp(for)35 b(all)i Fo(n)c Fn(\025)h Fo(n)599 1349 y Fk(1)p Fm(;h)698 1334 y Fp(,)j(for)f(all)g Fo(t)e Fn(2)g([)1259 1301 y Fm(s)1259 1359 y(j)t Fk(=1)1385 1334 y Fo(q)1429 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2062 y Fm(n)p Fk(=0)2268 2035 y Fp(\(1)30 b(+)e Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)2733 2002 y Fm(n)2782 2035 y Fo(t)p Fp(\)\)\))44 b(de\014nes)f(a)g (holomorphic)75 2148 y(function)30 b(on)g Fn(S)7 b Fp(.)40 b(Moreo)m(v)m(er,)33 b(b)m(y)d(construction)h(for)f(a)h(giv)m(en)g Fo(n)25 b Fn(\025)g Fp(0,)30 b(the)h(solutions)g(of)f(1)20 b(+)g Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)3536 2115 y Fm(n)3584 2148 y Fo(t)p Fp(\)\))26 b(=)f(0)75 2261 y(are)38 b(of)h(the)f(form)f Fo(t)h Fp(=)g Fo(q)956 2228 y Fl(\000)p Fm(n)1057 2261 y Fo(=n)1157 2275 y Fm(j)1194 2261 y Fp(,)i(1)e Fn(\024)g Fo(j)44 b Fn(\024)37 b Fo(s)p Fp(.)63 b(Moreo)m(v)m(er,)43 b(these)38 b(complex)h(n)m(um)m(b)s(ers)d(are)j(p)s(oles)f(for)f(the)75 2374 y(function)44 b(1)p Fo(=)p Fp(\(1)31 b(+)e Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)1002 2341 y Fm(n)1051 2374 y Fo(t)p Fp(\)\)\))45 b(b)s(ecause)f Fo(u)g Fp(is)g(a)h(rational)g (function.)82 b(So)44 b(that)h(the)g(function)f Fo(e)3594 2388 y Fm(q)3632 2374 y Fp(\()p Fo(t)p Fp(\))49 b(=)75 2487 y(1)p Fo(=)180 2419 y Fj(Q)267 2445 y Fk(+)p Fl(1)267 2514 y Fm(n)p Fk(=0)404 2487 y Fp(\(1)22 b(+)e Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)853 2454 y Fm(n)902 2487 y Fo(t)p Fp(\)\)\))32 b(is)f(holomorphic)h(on)f Fn(S)38 b Fp(outside)31 b(the)h(p)s(oin)m(ts)f Fo(q)2696 2454 y Fl(\000)p Fm(n)2798 2487 y Fo(=n)2898 2501 y Fm(j)2934 2487 y Fp(,)h(1)27 b Fn(\024)g Fo(j)32 b Fn(\024)27 b Fo(s)p Fp(,)k Fo(n)26 b Fn(\025)h Fp(0,)32 b(and)75 2600 y(is)h(meromophic)g(at)g(these)h(p)s(oin)m(ts.)48 b(As)33 b(in)f([8)q(],)i(page)f(72,)i(one)e(v)m(eri\014es)g(that)h Fo(e)2833 2614 y Fm(q)2871 2600 y Fp(\()p Fo(t)p Fp(\))f(is)g(formally) h(a)f(solution)75 2713 y(of)f(\(3\))g(b)m(y)g(plugging)f(the)h (expression)f(1)p Fo(=)1526 2645 y Fj(Q)1613 2671 y Fk(+)p Fl(1)1613 2740 y Fm(n)p Fk(=0)1750 2713 y Fp(\(1)22 b(+)f Fo(u)p Fp(\(1)p Fo(=)p Fp(\()p Fo(q)2200 2680 y Fm(n)2248 2713 y Fo(t)p Fp(\)\)\))32 b(in)f(\(3\).)45 b(The)31 b(part)h(b\))f(of)h(Prop)s(osition)f(1)75 2826 y(follo)m(ws.)3389 b Fa(2)75 3162 y Fq(3)135 b(A)39 b(Cauc)l(h)l(y)h(problem)g(in)g(a)g (class)g(of)g(en)l(tire)i(functions)d(with)h Fe(q)t Fd(\000)p Fq(exp)t(o-)277 3311 y(nen)l(tial)46 b(gro)l(wth)75 3514 y Fc(De\014nition)35 b(1)46 b Ff(L)-5 b(et)34 b Fo(q)s(;)15 b(T)8 b(;)15 b(X)41 b Ff(b)-5 b(e)34 b(r)-5 b(e)g(al)35 b(numb)-5 b(ers)35 b(such)f(that)h Fo(q)30 b(>)d Fp(1)p Ff(,)34 b Fo(T)8 b(;)15 b(X)35 b(>)27 b Fp(0)p Ff(.)45 b(We)34 b(de\014ne)g(a)g(ve)-5 b(ctor)34 b(sp)-5 b(ac)g(e)75 3627 y Fo(G)146 3594 y Fm(q)185 3627 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))38 b Ff(which)g(is)f(a)g(subsp)-5 b(ac)g(e)39 b(of)e(the)h(ve)-5 b(ctor)37 b(sp)-5 b(ac)g(e)39 b(of)e(entir)-5 b(e)38 b(functions)f(on)h Fg(C)3047 3594 y Fk(2)3123 3627 y Ff(denote)-5 b(d)39 b(by)e Fo(A)p Fp(\()p Fg(C)3752 3594 y Fk(2)3791 3627 y Fp(\))p Ff(.)75 3740 y(A)n(n)32 b(entir)-5 b(e)33 b(function)1516 3907 y Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)1927 3820 y Fj(X)1905 4018 y Fm(l)q(;\014)s Fl(\025)p Fk(0)2095 3907 y Fo(u)2147 3922 y Fm(l)q(;\014)2236 3907 y Fo(t)2269 3869 y Fm(l)2304 3845 y Fo(x)2356 3812 y Fm(\014)p 2304 3886 100 4 v 2313 3969 a Fo(\014)5 b Fp(!)75 4170 y Ff(b)-5 b(elongs)33 b(to)h Fo(G)567 4137 y Fm(q)605 4170 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(,)34 b(if)e(the)h(series)1597 4246 y Fj(X)1575 4444 y Fm(l)q(;\014)s Fl(\025)p Fk(0)1841 4270 y Fn(j)p Fo(u)1918 4285 y Fm(l)q(;\014)2007 4270 y Fn(j)p 1775 4312 324 4 v 1775 4399 a Fo(q)1819 4373 y Fm(P)10 b Fk(\()p Fm(l)q(;\014)s Fk(\))2017 4399 y Fo(\014)5 b Fp(!)2108 4333 y Fo(T)2174 4295 y Fm(l)2200 4333 y Fo(X)2282 4295 y Fm(\014)2329 4333 y Fo(;)75 4597 y Ff(c)-5 b(onver)g(ges,)33 b(wher)-5 b(e)34 b Fo(P)13 b Fp(\()p Fo(l)r(;)i(\014)5 b Fp(\))34 b Ff(denotes)f(the)g(p)-5 b(olynomial)1535 4801 y Fo(P)13 b Fp(\()p Fo(l)r(;)i(\014)5 b Fp(\))26 b(=)f Fo(l)r(\014)h Fn(\000)20 b Fo(\014)2176 4763 y Fk(2)2235 4801 y Fn(\000)g Fo(l)2355 4763 y Fk(2)75 5005 y Ff(and)34 b(we)e(also)i(de\014ne)f(a)g(weighte)-5 b(d)34 b Fo(L)1343 4972 y Fk(1)1415 5005 y Ff(norm)f(on)g Fo(G)1858 4972 y Fm(q)1897 5005 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))34 b Ff(as)1263 5259 y Fn(jj)p Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)1630 5273 y Fm(T)6 b(;X)1791 5259 y Fp(=)1909 5172 y Fj(X)1887 5370 y Fm(l)q(;\014)s Fl(\025)p Fk(0)2153 5196 y Fn(j)p Fo(u)2230 5211 y Fm(l)q(;\014)2319 5196 y Fn(j)p 2087 5238 V 2087 5325 a Fo(q)2131 5299 y Fm(P)k Fk(\()p Fm(l)q(;\014)s Fk(\))2329 5325 y Fo(\014)5 b Fp(!)2420 5259 y Fo(T)2486 5221 y Fm(l)2512 5259 y Fo(X)2594 5221 y Fm(\014)2641 5259 y Fo(:)75 5560 y Ff(One)32 b(c)-5 b(an)33 b(e)-5 b(asily)34 b(show)g(that)g Fp(\()p Fo(G)1201 5527 y Fm(q)1240 5560 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b Fn(jj)p Fo(:)p Fn(jj)1658 5574 y Fm(T)6 b(;X)1795 5560 y Fp(\))33 b Ff(is)f(a)h(Banach)h(sp)-5 b(ac)g(e.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Fp(6)75 399 y Fc(Remark:)48 b Fp(Let)34 b Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))35 b(b)s(e)e(in)h Fo(G)1290 366 y Fm(q)1328 399 y Fp(\()p Fo(T)1416 413 y Fk(0)1456 399 y Fo(;)15 b(X)1571 413 y Fk(0)1611 399 y Fp(\))34 b(for)g(giv)m(en)h Fo(T)2117 413 y Fk(0)2156 399 y Fo(;)15 b(X)2271 413 y Fk(0)2342 399 y Fo(>)31 b Fp(0.)52 b(Then,)33 b Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))35 b(also)g(b)s(elong)f(to)g(the)75 511 y(spaces)41 b Fo(G)435 478 y Fm(q)474 511 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))42 b(for)f(all)h Fo(X)50 b Fn(\024)43 b Fo(X)1370 525 y Fk(0)1451 511 y Fp(and)d Fo(T)56 b Fn(\024)43 b Fo(T)1914 525 y Fk(0)1953 511 y Fp(.)73 b(Moreo)m(v)m(er,)46 b(the)41 b(maps)g Fo(X)50 b Fn(7!)43 b(jj)p Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)3531 525 y Fm(T)6 b(;X)3708 511 y Fp(and)75 624 y Fo(T)38 b Fn(7!)25 b(jj)p Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)649 638 y Fm(T)6 b(;X)815 624 y Fp(are)31 b(increasing)g(functions)f(from)g([0)p Fo(;)15 b(X)2184 638 y Fk(0)2225 624 y Fp(])30 b(\(resp.)41 b([0)p Fo(;)15 b(T)2707 638 y Fk(0)2747 624 y Fp(]\))31 b(in)m(to)h Fg(R)3089 638 y Fk(+)3147 624 y Fp(.)75 787 y(W)-8 b(e)32 b(de\014ne)d(the)i(in)m(tegration)h(op)s(erator)f Fo(@)1533 754 y Fl(\000)p Fk(1)1528 810 y Fm(x)1653 787 y Fp(:)25 b Fo(A)p Fp(\()p Fg(C)1872 754 y Fk(2)1912 787 y Fp(\))g Fn(!)h Fo(A)p Fp(\()p Fg(C)2258 754 y Fk(2)2297 787 y Fp(\))31 b(as)1418 1035 y Fo(@)1471 998 y Fl(\000)p Fk(1)1466 1058 y Fm(x)1566 1035 y Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(:=)1980 912 y Fj(Z)2071 938 y Fm(x)2031 1118 y Fk(0)2130 1035 y Fo(U)10 b Fp(\()p Fo(t;)15 b(\020)7 b Fp(\))p Fo(d\020)g(:)75 1294 y Fc(De\014nition)35 b(2)46 b Ff(F)-7 b(or)30 b(any)g(p)-5 b(olynomial)32 b Fo(a)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)1817 1226 y Fj(P)1913 1321 y Fm(\013;\026)p Fl(\025)p Fk(0)2130 1294 y Fo(a)2178 1308 y Fm(\013;\026)2289 1294 y Fo(t)2322 1261 y Fm(\013)2371 1294 y Fo(x)2423 1261 y Fm(\026)2495 1294 y Fn(2)f Fg(C)p Fp([)p Fo(t;)15 b(x)p Fp(])p Ff(,)30 b(we)f(denote)h(by)f Fn(j)p Fo(a)p Fn(j)p Fp(\()p Fo(t;)15 b(x)p Fp(\))30 b Ff(the)75 1407 y(p)-5 b(olynomial)36 b Fn(j)p Fo(a)p Fn(j)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)948 1339 y Fj(P)1044 1434 y Fm(\013;\026)p Fl(\025)p Fk(0)1261 1407 y Fn(j)p Fo(a)1334 1421 y Fm(\013;\026)1446 1407 y Fn(j)p Fo(t)1504 1374 y Fm(\013)1554 1407 y Fo(x)1606 1374 y Fm(\026)1652 1407 y Ff(.)75 1619 y Fc(De\014nition)35 b(3)46 b Ff(L)-5 b(et)33 b Fo(h)870 1633 y Fk(2)910 1619 y Fo(;)15 b(m)1030 1633 y Fk(1)1069 1619 y Fo(;)g(m)1189 1633 y Fk(2)1254 1619 y Fn(\025)25 b Fp(0)33 b Ff(b)-5 b(e)33 b(non)g(ne)-5 b(gative)33 b(inte)-5 b(gers.)42 b(L)-5 b(et)33 b(the)g(set)514 1824 y Fn(P)577 1839 y Fm(h)618 1848 y Fi(2)652 1839 y Fm(;m)734 1848 y Fi(1)769 1839 y Fm(;m)851 1848 y Fi(2)915 1824 y Fp(=)25 b Fn(f)p Fp(\()p Fo(\013;)15 b(\026)p Fp(\))27 b Fn(2)e Fg(N)1458 1786 y Fk(2)1587 1824 y Fp(:)91 b Fo(h)1755 1838 y Fk(2)1815 1824 y Fp(+)20 b Fo(\026)25 b Fn(\025)g Fp(2)p Fo(\013)c Fp(+)f Fo(m)2377 1838 y Fk(1)2481 1824 y Fo(;)80 b(\013)21 b Fp(+)f Fo(m)2836 1838 y Fk(2)2901 1824 y Fn(\025)k Fp(2\()p Fo(h)3128 1838 y Fk(2)3189 1824 y Fp(+)c Fo(\026)p Fp(\))p Fn(g)75 2028 y Ff(assume)-5 b(d)35 b(not)e(to)h(b)-5 b(e)33 b(empty.)44 b(A)33 b(p)-5 b(olynomial)36 b Fo(a)p Fp(\()p Fo(t;)15 b(x)p Fp(\))27 b Fn(2)f Fg(C)p Fp([)p Fo(t;)15 b(x)p Fp(])34 b Ff(is)f(said)h(of)f Fn(P)2776 2043 y Fm(h)2817 2052 y Fi(2)2852 2043 y Fm(;m)2934 2052 y Fi(1)2968 2043 y Fm(;m)3050 2052 y Fi(2)3089 2028 y Fn(\000)p Ff(typ)-5 b(e)34 b(if)e(it)i(is)f(of)g(the)75 2141 y(form)1317 2262 y Fo(a)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)1891 2176 y Fj(X)1682 2377 y Fk(\()p Fm(\013;\026)p Fk(\))p Fl(2P)1939 2389 y Fh(h)1975 2404 y Fi(2)2012 2389 y Fh(;m)2085 2404 y Fi(1)2120 2389 y Fh(;m)2193 2404 y Fi(2)2247 2262 y Fo(a)2295 2276 y Fm(\013;\026)2406 2262 y Fo(t)2439 2225 y Fm(\013)2489 2262 y Fo(x)2541 2225 y Fm(\026)2587 2262 y Fo(:)75 2589 y Fc(Remark:)38 b Fp(If)25 b(\()p Fo(\013;)15 b(\026)p Fp(\))27 b Fn(2)e(P)989 2604 y Fm(h)1030 2613 y Fi(2)1064 2604 y Fm(;m)1146 2613 y Fi(1)1181 2604 y Fm(;m)1263 2613 y Fi(2)1327 2589 y Fp(then)g(w)m(e)h(get)g(0)g Fn(\024)f Fp(3)p Fo(\013)h Fn(\024)f Fo(m)2276 2603 y Fk(2)2325 2589 y Fn(\000)10 b Fp(2)p Fo(m)2531 2603 y Fk(1)2596 2589 y Fp(and)25 b(0)g Fn(\024)g Fp(3)p Fo(\026)h Fn(\024)f(\000)p Fp(3)p Fo(h)3324 2603 y Fk(2)3374 2589 y Fp(+)10 b(2)p Fo(m)3580 2603 y Fk(2)3629 2589 y Fn(\000)g Fo(m)3790 2603 y Fk(1)3829 2589 y Fp(.)75 2702 y(In)30 b(particular,)h(the)f(set)h Fn(P)995 2717 y Fm(h)1036 2726 y Fi(2)1071 2717 y Fm(;m)1153 2726 y Fi(1)1187 2717 y Fm(;m)1269 2726 y Fi(2)1338 2702 y Fp(is)g(b)s(ounded)d(in)i Fg(N)1973 2669 y Fk(2)2012 2702 y Fp(.)75 2915 y Fc(Lemma)35 b(2)45 b Ff(L)-5 b(et)38 b Fo(h)757 2929 y Fk(2)796 2915 y Fo(;)15 b(m)916 2929 y Fk(1)956 2915 y Fo(;)g(m)1076 2929 y Fk(2)1148 2915 y Fn(\025)33 b Fp(0)k Ff(b)-5 b(e)37 b(non)g(ne)-5 b(gative)37 b(inte)-5 b(gers.)55 b(Assume)36 b(that)i(the)g(set)e Fn(P)3275 2930 y Fm(h)3316 2939 y Fi(2)3351 2930 y Fm(;m)3433 2939 y Fi(1)3468 2930 y Fm(;m)3550 2939 y Fi(2)3625 2915 y Ff(is)h(not)75 3027 y(empty.)43 b(Then,)32 b(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)34 b Fo(C)d(>)25 b Fp(0)33 b Ff(\(dep)-5 b(ending)34 b(on)f Fo(h)2474 3041 y Fk(2)2513 3027 y Fo(;)15 b(m)2633 3041 y Fk(1)2673 3027 y Fo(;)g(m)2793 3041 y Fk(2)2833 3027 y Ff(\))32 b(such)h(that)g(for)g(al)5 b(l)33 b(p)-5 b(olyno-)75 3140 y(mials)34 b Fo(a)p Fp(\()p Fo(t;)15 b(x)p Fp(\))33 b Ff(of)g Fn(P)763 3155 y Fm(h)804 3164 y Fi(2)838 3155 y Fm(;m)920 3164 y Fi(1)955 3155 y Fm(;m)1037 3164 y Fi(2)1076 3140 y Fn(\000)p Ff(typ)-5 b(e,)33 b(one)g(has)685 3345 y Fn(jj)p Fo(a)p Fp(\()p Fo(t;)15 b(x)p Fp(\)\()p Fo(@)1066 3307 y Fl(\000)p Fm(h)1162 3316 y Fi(2)1061 3367 y Fm(x)1202 3345 y Fo(U)10 b Fp(\)\()p Fo(q)1388 3307 y Fm(m)1450 3316 y Fi(1)1490 3345 y Fo(t;)15 b(q)1607 3307 y Fl(\000)p Fm(m)1724 3316 y Fi(2)1763 3345 y Fo(x)p Fp(\))p Fn(jj)1900 3359 y Fm(T)6 b(;X)2060 3345 y Fn(\024)25 b Fo(C)7 b Fn(j)p Fo(a)p Fn(j)p Fp(\()p Fo(T)h(;)15 b(X)7 b Fp(\))p Fo(X)2661 3307 y Fm(h)2702 3316 y Fi(2)2742 3345 y Fn(jj)p Fo(U)j Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)3109 3359 y Fm(T)6 b(;X)75 3549 y Ff(for)33 b(al)5 b(l)33 b Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)800 3516 y Fm(q)839 3549 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)75 3811 y Fc(Pro)s(of)31 b Fp(F)-8 b(or)31 b Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)972 3778 y Fm(q)1011 3811 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))31 b(w)m(e)g(ha)m(v)m(e)274 4074 y Fo(a)322 4088 y Fm(\013;\026)434 4074 y Fo(t)467 4037 y Fm(\013)516 4074 y Fo(x)568 4037 y Fm(\026)615 4074 y Fp(\()p Fo(@)703 4037 y Fl(\000)p Fm(h)799 4046 y Fi(2)698 4097 y Fm(x)838 4074 y Fo(U)10 b Fp(\)\()p Fo(q)1024 4037 y Fm(m)1086 4046 y Fi(1)1126 4074 y Fo(t;)15 b(q)1243 4037 y Fl(\000)p Fm(m)1360 4046 y Fi(2)1399 4074 y Fo(x)p Fp(\))25 b(=)g Fo(a)1655 4088 y Fm(\013;\026)1782 3988 y Fj(X)1805 4185 y Fm(l)q(;\014)2057 4013 y Fo(\014)5 b Fp(!)p 1938 4053 319 4 v 1938 4136 a(\()p Fo(\014)26 b Fn(\000)20 b Fo(\026)p Fp(\)!)2266 4074 y Fo(u)2318 4089 y Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(h)2599 4098 y Fi(2)2633 4089 y Fl(\000)p Fm(\026)2735 4074 y Fo(q)2779 4037 y Fm(m)2841 4046 y Fi(1)2876 4037 y Fk(\()p Fm(l)q Fl(\000)p Fm(\013)p Fk(\))3057 4074 y Fo(q)3101 4037 y Fl(\000)p Fm(m)3218 4046 y Fi(2)3253 4037 y Fk(\()p Fm(\014)s Fl(\000)p Fm(\026)p Fk(\))3452 4074 y Fo(t)3485 4037 y Fm(l)3521 4013 y Fo(x)3573 3980 y Fm(\014)p 3521 4053 100 4 v 3530 4136 a Fo(\014)5 b Fp(!)3630 4074 y Fo(:)75 4370 y Fp(By)31 b(de\014nition,)f(w)m(e)h(ha)m(v)m(e)158 4599 y Fn(jj)p Fo(a)256 4613 y Fm(\013;\026)368 4599 y Fo(t)401 4562 y Fm(\013)450 4599 y Fo(x)502 4562 y Fm(\026)549 4599 y Fp(\()p Fo(@)637 4562 y Fl(\000)p Fm(h)733 4571 y Fi(2)632 4622 y Fm(x)772 4599 y Fo(U)10 b Fp(\)\()p Fo(q)958 4562 y Fm(m)1020 4571 y Fi(1)1060 4599 y Fo(t;)15 b(q)1177 4562 y Fl(\000)p Fm(m)1294 4571 y Fi(2)1333 4599 y Fo(x)p Fp(\))p Fn(jj)1470 4613 y Fm(T)6 b(;X)609 4812 y Fp(=)25 b Fn(j)p Fo(a)778 4826 y Fm(\013;\026)890 4812 y Fn(j)p Fo(T)981 4775 y Fm(\013)1030 4812 y Fo(X)1112 4775 y Fm(\026)p Fk(+)p Fm(h)1250 4784 y Fi(2)1304 4726 y Fj(X)1328 4924 y Fm(l)q(;\014)1461 4749 y Fn(j)p Fo(u)1538 4764 y Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(h)1819 4773 y Fi(2)1853 4764 y Fl(\000)p Fm(\026)1954 4749 y Fn(j)p Fp(\()p Fo(\014)5 b Fp(!)p Fo(=)p Fp(\()p Fo(\014)28 b Fn(\000)20 b Fo(\026)p Fp(\)!\))p Fo(q)2539 4716 y Fm(m)2601 4725 y Fi(1)2636 4716 y Fk(\()p Fm(l)q Fl(\000)p Fm(\013)p Fk(\))2817 4749 y Fo(q)2861 4716 y Fl(\000)p Fm(m)2978 4725 y Fi(2)3013 4716 y Fk(\()p Fm(\014)s Fl(\000)p Fm(\026)p Fk(\))p 1461 4791 1752 4 v 2175 4879 a Fo(q)2219 4852 y Fm(P)10 b Fk(\()p Fm(l)q(;\014)s Fk(\))2417 4879 y Fo(\014)5 b Fp(!)3222 4812 y Fo(T)3288 4775 y Fm(l)q Fl(\000)p Fm(\013)3414 4812 y Fo(X)3496 4775 y Fm(\014)s Fl(\000)p Fm(h)3635 4784 y Fi(2)3670 4775 y Fl(\000)p Fm(\026)75 5108 y Fp(So)30 b(that)158 5338 y Fn(jj)p Fo(a)256 5352 y Fm(\013;\026)368 5338 y Fo(t)401 5300 y Fm(\013)450 5338 y Fo(x)502 5300 y Fm(\026)549 5338 y Fp(\()p Fo(@)637 5300 y Fl(\000)p Fm(h)733 5309 y Fi(2)632 5360 y Fm(x)772 5338 y Fo(U)10 b Fp(\)\()p Fo(q)958 5300 y Fm(m)1020 5309 y Fi(1)1060 5338 y Fo(t;)15 b(q)1177 5300 y Fl(\000)p Fm(m)1294 5309 y Fi(2)1333 5338 y Fo(x)p Fp(\))p Fn(jj)1470 5352 y Fm(T)6 b(;X)1005 5536 y Fp(=)25 b Fn(j)p Fo(a)1174 5550 y Fm(\013;\026)1286 5536 y Fn(j)p Fo(T)1377 5498 y Fm(\013)1427 5536 y Fo(X)1509 5498 y Fm(\026)p Fk(+)p Fm(h)1647 5507 y Fi(2)1701 5450 y Fj(X)1724 5647 y Fm(l)q(;\014)1848 5536 y Fo(A)p Fp(\()p Fo(l)r(;)15 b(\014)5 b Fp(\))2408 5473 y Fn(j)p Fo(u)2485 5488 y Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(h)2766 5497 y Fi(2)2800 5488 y Fl(\000)p Fm(\026)2901 5473 y Fn(j)p 2121 5515 1091 4 v 2121 5602 a Fo(q)2165 5576 y Fm(P)10 b Fk(\()p Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(h)2528 5585 y Fi(2)2563 5576 y Fl(\000)p Fm(\026)p Fk(\))2691 5602 y Fp(\()p Fo(\014)26 b Fn(\000)20 b Fo(h)2946 5616 y Fk(2)3006 5602 y Fn(\000)g Fo(\026)p Fp(\)!)3222 5536 y Fo(T)3288 5498 y Fm(l)q Fl(\000)p Fm(\013)3414 5536 y Fo(X)3496 5498 y Fm(\014)s Fl(\000)p Fm(h)3635 5507 y Fi(2)3670 5498 y Fl(\000)p Fm(\026)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fp(7)75 399 y(where)816 536 y Fo(A)p Fp(\()p Fo(l)r(;)15 b(\014)5 b Fp(\))26 b(=)1884 475 y(\()p Fo(\014)g Fn(\000)20 b Fo(h)2139 489 y Fk(2)2199 475 y Fn(\000)g Fo(\026)p Fp(\)!)p 1211 516 1868 4 v 1211 603 a(\()p Fo(\014)26 b Fn(\000)20 b Fo(\026)p Fp(\)!)p Fo(q)1573 576 y Fk(\()p Fm(P)10 b Fk(\()p Fm(l)q(;\014)s Fk(\))p Fl(\000)p Fm(P)g Fk(\()p Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(h)2212 585 y Fi(2)2246 576 y Fl(\000)p Fm(\026)p Fk(\))p Fl(\000)p Fm(m)2487 585 y Fi(1)2523 576 y Fk(\()p Fm(l)q Fl(\000)p Fm(\013)p Fk(\)+)p Fm(m)2816 585 y Fi(2)2852 576 y Fk(\()p Fm(\014)s Fl(\000)p Fm(\026)p Fk(\)\))3089 536 y Fo(:)75 751 y Fp(By)31 b(construction,)g(w)m(e)g(ha)m(v)m(e)158 980 y Fo(P)13 b Fp(\()p Fo(l)r(;)i(\014)5 b Fp(\))21 b Fn(\000)f Fo(P)13 b Fp(\()p Fo(l)23 b Fn(\000)d Fo(\013;)15 b(\014)26 b Fn(\000)20 b Fo(h)1101 994 y Fk(2)1161 980 y Fn(\000)g Fo(\026)p Fp(\))g Fn(\000)g Fo(m)1533 994 y Fk(1)1572 980 y Fp(\()p Fo(l)j Fn(\000)d Fo(\013)p Fp(\))h(+)f Fo(m)2033 994 y Fk(2)2072 980 y Fp(\()p Fo(\014)26 b Fn(\000)20 b Fo(\026)p Fp(\))100 1127 y(=)25 b Fo(l)r Fp(\()p Fo(h)312 1141 y Fk(2)365 1127 y Fp(+)13 b Fo(\026)g Fn(\000)g Fp(2)p Fo(\013)g Fn(\000)g Fo(m)881 1141 y Fk(1)920 1127 y Fp(\))g(+)g Fo(\014)5 b Fp(\()p Fo(\013)13 b Fn(\000)g Fp(2)p Fo(h)1395 1141 y Fk(2)1448 1127 y Fn(\000)g Fp(2)p Fo(\026)g Fp(+)g Fo(m)1809 1141 y Fk(2)1848 1127 y Fp(\))g Fn(\000)g Fo(\013h)2090 1141 y Fk(2)2143 1127 y Fn(\000)g Fo(\013\026)g Fp(+)g Fo(h)2489 1090 y Fk(2)2489 1150 y(2)2541 1127 y Fp(+)g(2)p Fo(h)2722 1141 y Fk(2)2762 1127 y Fo(\026)g Fp(+)g Fo(\026)2969 1090 y Fk(2)3020 1127 y Fp(+)g Fo(\013)3162 1090 y Fk(2)3214 1127 y Fp(+)g Fo(m)3378 1141 y Fk(1)3417 1127 y Fo(\013)g Fn(\000)g Fo(m)3652 1141 y Fk(2)3691 1127 y Fo(\026:)75 1332 y Fp(So)29 b(that)i(there)e(exists)i(a)f(constan)m(t)h Fo(C)g(>)25 b Fp(0)30 b(\(dep)s(ending)f(on)g Fo(h)2236 1346 y Fk(2)2276 1332 y Fo(;)15 b(m)2396 1346 y Fk(1)2436 1332 y Fo(;)g(m)2556 1346 y Fk(2)2595 1332 y Fp(\))30 b(suc)m(h)f(that)i(if)e(\()p Fo(\013;)15 b(\026)p Fp(\))27 b Fn(2)e(P)3542 1347 y Fm(h)3583 1356 y Fi(2)3617 1347 y Fm(;m)3699 1356 y Fi(1)3734 1347 y Fm(;m)3816 1356 y Fi(2)75 1445 y Fp(then)30 b Fo(A)p Fp(\()p Fo(l)r(;)15 b(\014)5 b Fp(\))27 b Fn(\024)e Fo(C)7 b Fp(,)30 b(for)g(all)h Fo(l)r(;)15 b(\014)5 b Fp(.)41 b(This)30 b(giv)m(es)i(the)e(lemma.)1631 b Fa(2)75 1707 y Fc(Lemma)35 b(3)45 b Ff(L)-5 b(et)33 b Fo(\016)740 1721 y Fk(1)780 1707 y Fo(;)15 b(\016)860 1721 y Fk(2)925 1707 y Fo(>)25 b Fp(0)33 b Ff(b)-5 b(e)32 b(p)-5 b(ositive)34 b(r)-5 b(e)g(al)34 b(numb)-5 b(ers)33 b(and)g(let)g Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))33 b Ff(b)-5 b(e)32 b(in)g Fo(G)2969 1674 y Fm(q)3008 1707 y Fp(\()p Fo(T)3096 1721 y Fk(0)3136 1707 y Fo(;)15 b(X)3251 1721 y Fk(0)3291 1707 y Fp(\))p Ff(.)42 b(Then,)33 b(ther)-5 b(e)75 1820 y(exist)37 b Fo(X)r(;)15 b(T)47 b(>)32 b Fp(0)38 b Ff(smal)5 b(l)37 b(enough)h(\(dep)-5 b(ending)38 b(on)f Fo(\016)1891 1834 y Fk(1)1931 1820 y Fo(;)15 b(\016)2011 1834 y Fk(2)2051 1820 y Ff(\))37 b(such)g(that)i(the)e(function)g Fo(U)10 b Fp(\()p Fo(\016)3191 1834 y Fk(1)3231 1820 y Fo(t;)3327 1784 y Fm(x)p 3314 1799 66 4 v 3314 1851 a(\016)3345 1860 y Fi(2)3390 1820 y Fp(\))37 b Ff(b)-5 b(elongs)38 b(to)75 1933 y Fo(G)146 1900 y Fm(q)185 1933 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)75 2195 y Fc(Pro)s(of)31 b Fp(Let)g Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)972 2162 y Fm(q)1011 2195 y Fp(\()p Fo(T)1099 2209 y Fk(0)1139 2195 y Fo(;)15 b(X)1254 2209 y Fk(0)1294 2195 y Fp(\).)41 b(W)-8 b(e)31 b(ha)m(v)m(e)1296 2458 y Fo(U)10 b Fp(\()p Fo(\016)1443 2472 y Fk(1)1483 2458 y Fo(t;)1580 2397 y(x)p 1566 2437 80 4 v 1566 2520 a(\016)1606 2534 y Fk(2)1656 2458 y Fp(\))26 b(=)1834 2372 y Fj(X)1812 2569 y Fm(l)q(;\014)s Fl(\025)p Fk(0)2002 2458 y Fo(u)2054 2473 y Fm(l)q(;\014)2143 2458 y Fo(\016)2183 2472 y Fk(1)2223 2420 y Fm(l)2249 2458 y Fo(\016)2289 2472 y Fk(2)2329 2420 y Fl(\000)p Fm(\014)2431 2458 y Fo(t)2464 2420 y Fm(l)2499 2397 y Fo(x)2551 2364 y Fm(\014)p 2499 2437 100 4 v 2508 2520 a Fo(\014)5 b Fp(!)2609 2458 y Fo(:)75 2754 y Fp(By)31 b(de\014nition,)f(w)m(e)h(ha)m(v)m(e)1051 3003 y Fn(jj)p Fo(U)10 b Fp(\()p Fo(\016)1248 3017 y Fk(1)1289 3003 y Fo(t;)1386 2941 y(x)p 1372 2982 80 4 v 1372 3065 a(\016)1412 3079 y Fk(2)1462 3003 y Fp(\))p Fn(jj)1547 3017 y Fm(T)c(;X)1707 3003 y Fp(=)1803 2916 y Fj(X)1826 3114 y Fm(l)q(;\014)2025 2940 y Fn(j)p Fo(u)2102 2955 y Fm(l)q(;\014)2191 2940 y Fn(j)p 1959 2982 324 4 v 1959 3069 a Fo(q)2003 3043 y Fm(P)k Fk(\()p Fm(l)q(;\014)s Fk(\))2201 3069 y Fo(\014)5 b Fp(!)2293 3003 y(\()p Fo(\016)2368 3017 y Fk(1)2408 3003 y Fo(T)13 b Fp(\))2509 2965 y Fm(l)2535 3003 y Fp(\()p Fo(X=\016)2730 3017 y Fk(2)2771 3003 y Fp(\))2806 2965 y Fm(\014)2853 3003 y Fo(:)75 3304 y Fp(Due)22 b(to)h(the)g(fact)g(that)g Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)1319 3271 y Fm(q)1357 3304 y Fp(\()p Fo(T)1445 3318 y Fk(0)1485 3304 y Fo(;)15 b(X)1600 3318 y Fk(0)1640 3304 y Fp(\),)25 b(this)d(latter)h(series)g (con)m(v)m(erges)h(if)e Fo(X=\016)3003 3318 y Fk(2)3068 3304 y Fn(\024)j Fo(X)3239 3318 y Fk(0)3301 3304 y Fp(and)d Fo(\016)3510 3318 y Fk(1)3550 3304 y Fo(T)38 b Fn(\024)25 b Fo(T)3790 3318 y Fk(0)3829 3304 y Fp(.)75 3417 y(So)30 b(that,)h(if)g Fo(X=\016)667 3431 y Fk(2)732 3417 y Fn(\024)25 b Fo(X)903 3431 y Fk(0)973 3417 y Fp(and)30 b Fo(\016)1190 3431 y Fk(1)1230 3417 y Fo(T)38 b Fn(\024)25 b Fo(T)1470 3431 y Fk(0)1509 3417 y Fp(,)31 b(then)f Fo(U)10 b Fp(\()p Fo(\016)1919 3431 y Fk(1)1959 3417 y Fo(t;)2055 3381 y Fm(x)p 2042 3396 66 4 v 2042 3448 a(\016)2073 3457 y Fi(2)2118 3417 y Fp(\))26 b Fn(2)f Fo(G)2336 3384 y Fm(q)2374 3417 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\).)1104 b Fa(2)75 3679 y Fc(Lemma)35 b(4)45 b Ff(L)-5 b(et)38 b Fo(\013;)15 b(\026)34 b Fn(2)e Fg(N)37 b Ff(and)h(let)f Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))34 b Fn(2)f Fo(G)1867 3646 y Fm(q)1905 3679 y Fp(\()p Fo(T)1993 3693 y Fk(0)2033 3679 y Fo(;)15 b(X)2148 3693 y Fk(0)2189 3679 y Fp(\))p Ff(.)55 b(Then,)38 b(ther)-5 b(e)38 b(exist)f Fo(T)8 b(;)15 b(X)41 b(>)33 b Fp(0)k Ff(\(dep)-5 b(ending)75 3792 y(on)33 b Fo(\013;)15 b(\026)p Ff(\))33 b(smal)5 b(l)34 b(enough)f(such)g(that)h(the)f(function)f Fo(t)1913 3759 y Fm(\013)1963 3792 y Fo(x)2015 3759 y Fm(\026)2061 3792 y Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))34 b Ff(b)-5 b(elongs)33 b(to)g Fo(G)2853 3759 y Fm(q)2892 3792 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)75 4054 y Fc(Pro)s(of)37 b Fp(Let)f Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))36 b Fn(2)e Fo(G)1002 4021 y Fm(q)1040 4054 y Fp(\()p Fo(T)1128 4068 y Fk(0)1168 4054 y Fo(;)15 b(X)1283 4068 y Fk(0)1323 4054 y Fp(\).)58 b(F)-8 b(rom)36 b(the)g(pro)s(of)f (of)i(Lemma)f(2,)h(w)m(e)g(get)g(a)f(constan)m(t)h Fo(C)k(>)34 b Fp(0)j(suc)m(h)75 4167 y(that)249 4398 y Fn(jj)p Fo(t)332 4361 y Fm(\013)382 4398 y Fo(x)434 4361 y Fm(\026)481 4398 y Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)798 4412 y Fm(T)6 b(;X)958 4398 y Fn(\024)25 b Fo(C)7 b Fp(\()p Fo(q)1205 4361 y Fk(2)p Fm(\013)1290 4398 y Fo(T)13 b Fp(\))1391 4361 y Fm(\013)1440 4398 y Fp(\()p Fo(q)1519 4361 y Fk(2)p Fm(\026)1601 4398 y Fo(X)7 b Fp(\))1718 4361 y Fm(\026)1766 4398 y Fo(q)1810 4361 y Fm(\013\026)1917 4312 y Fj(X)1940 4509 y Fm(l)q(;\014)2257 4335 y Fn(j)p Fo(u)2334 4350 y Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(\026)2621 4335 y Fn(j)p 2073 4377 758 4 v 2073 4464 a Fo(q)2117 4438 y Fm(P)j Fk(\()p Fm(l)q Fl(\000)p Fm(\013;\014)s Fl(\000)p Fm(\026)p Fk(\))2512 4464 y Fp(\()p Fo(\014)26 b Fn(\000)20 b Fo(\026)p Fp(\)!)2840 4398 y(\()p Fo(q)2919 4361 y Fk(2)p Fm(\013)3004 4398 y Fo(T)13 b Fp(\))3105 4361 y Fm(l)q Fl(\000)p Fm(\013)3232 4398 y Fp(\()p Fo(q)3311 4361 y Fk(2)p Fm(\026)3393 4398 y Fo(X)7 b Fp(\))3510 4361 y Fm(\014)s Fl(\000)p Fm(\026)3655 4398 y Fo(:)75 4699 y Fp(Due)22 b(to)h(the)g(fact)g(that)g Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)1319 4666 y Fm(q)1357 4699 y Fp(\()p Fo(T)1445 4713 y Fk(0)1485 4699 y Fo(;)15 b(X)1600 4713 y Fk(0)1640 4699 y Fp(\),)25 b(the)d(series)g(on)h(the)f(righ)m(t)h(hand)e(side)h(of)g(the)h(latter) g(inequalit)m(y)75 4812 y(is)30 b(con)m(v)m(ergen)m(t,)j(whic)m(h)e (yields)f Fo(t)1195 4779 y Fm(\013)1244 4812 y Fo(x)1296 4779 y Fm(\026)1343 4812 y Fo(U)10 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)1793 4779 y Fm(q)1831 4812 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\),)32 b(if)f Fo(q)2269 4779 y Fk(2)p Fm(\026)2350 4812 y Fo(X)i Fn(\024)25 b Fo(X)2629 4826 y Fk(0)2699 4812 y Fp(and)30 b Fo(q)2920 4779 y Fk(2)p Fm(\013)3004 4812 y Fo(T)38 b Fn(\024)25 b Fo(T)3244 4826 y Fk(0)3284 4812 y Fp(.)447 b Fa(2)216 4975 y Fp(Let)31 b Fn(D)i Fp(b)s(e)d(the)g(linear)h(op)s(erator)g(from) f Fo(A)p Fp(\()p Fg(C)1766 4942 y Fk(2)1805 4975 y Fp(\))h(in)m(to)g Fo(A)p Fp(\()p Fg(C)2224 4942 y Fk(2)2264 4975 y Fp(\))g(de\014ned)e (as)812 5265 y Fn(D)s Fp(\()p Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\)\))27 b(=)e Fo(@)1378 5228 y Fm(S)1373 5288 y(x)1429 5265 y Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))21 b Fn(\000)1788 5152 y Fm(S)t Fl(\000)p Fk(1)1791 5179 y Fj(X)1792 5377 y Fm(k)r Fk(=0)1940 5265 y Fo(a)1988 5280 y Fm(k)2031 5265 y Fp(\()p Fo(t;)15 b(x)p Fp(\)\()p Fo(@)2314 5228 y Fm(k)2309 5288 y(x)2358 5265 y Fo(u)p Fp(\)\()p Fo(q)2524 5228 y Fm(m)2586 5240 y Fi(0)p Fh(;k)2679 5265 y Fo(t;)g(xq)2848 5228 y Fl(\000)p Fm(m)2965 5240 y Fi(1)p Fh(;k)3057 5265 y Fp(\))p Fo(;)75 5554 y Fp(where)31 b Fo(S)g Fn(\025)c Fp(1,)32 b Fo(m)706 5569 y Fk(0)p Fm(;k)803 5554 y Fo(;)15 b(m)923 5569 y Fk(1)p Fm(;k)1048 5554 y Fn(\025)26 b Fp(0)32 b(are)f(in)m(tegers)i(and)d Fo(a)1937 5569 y Fm(k)1980 5554 y Fp(\()p Fo(t;)15 b(x)p Fp(\))27 b Fn(2)g Fg(C)p Fp([)p Fo(t;)15 b(x)p Fp(])31 b(for)g(0)c Fn(\024)g Fo(k)i Fn(\024)e Fo(S)f Fn(\000)20 b Fp(1.)44 b(W)-8 b(e)32 b(mak)m(e)g(the)75 5666 y(follo)m(wing)g (assumption.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Fp(8)75 399 y Fc(Assumption)36 b(\(A\))p Fp(:)75 511 y Fc(i\))30 b Fp(The)g(set)h Fn(P)567 526 y Fm(S)t Fl(\000)p Fm(k)r(;m)790 538 y Fi(0)p Fh(;k)877 526 y Fm(;m)959 538 y Fi(1)p Fh(;k)1080 511 y Fp(is)g(not)g(empt)m(y)-8 b(,)75 624 y Fc(ii\))30 b Fp(The)g(p)s(olynomial)h Fo(a)909 639 y Fm(k)951 624 y Fp(\()p Fo(t;)15 b(x)p Fp(\))32 b(is)e(of)g Fn(P)1435 639 y Fm(S)t Fl(\000)p Fm(k)r(;m)1658 651 y Fi(0)p Fh(;k)1745 639 y Fm(;m)1827 651 y Fi(1)p Fh(;k)1919 624 y Fn(\000)p Fp(t)m(yp)s(e,)75 787 y(for)g(all)h(0)26 b Fn(\024)f Fo(k)k Fn(\024)24 b Fo(S)i Fn(\000)20 b Fp(1.)75 950 y(W)-8 b(e)32 b(consider)e(the)g(follo)m(wing)i(op)s(erator)f(from) f Fo(A)p Fp(\()p Fg(C)1877 917 y Fk(2)1916 950 y Fp(\))h(in)m(to)g Fo(A)p Fp(\()p Fg(C)2335 917 y Fk(2)2375 950 y Fp(\),)1603 1154 y Fn(D)23 b(\016)e Fo(@)1815 1116 y Fl(\000)p Fm(S)1810 1176 y(x)1946 1154 y Fp(=)k(id)20 b Fn(\000)g(A)p Fo(;)75 1358 y Fp(where)938 1535 y Fn(A)p Fp(\()p Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\)\))27 b(=)1451 1421 y Fm(S)t Fl(\000)p Fk(1)1454 1448 y Fj(X)1455 1646 y Fm(k)r Fk(=0)1603 1535 y Fo(a)1651 1550 y Fm(k)1693 1535 y Fp(\()p Fo(t;)15 b(x)p Fp(\)\()p Fo(@)1976 1497 y Fl(\000)p Fk(\()p Fm(S)t Fl(\000)p Fm(k)r Fk(\))1971 1557 y Fm(x)2232 1535 y Fo(u)p Fp(\)\()p Fo(q)2398 1497 y Fm(m)2460 1509 y Fi(0)p Fh(;k)2553 1535 y Fo(t;)g(xq)2722 1497 y Fl(\000)p Fm(m)2839 1509 y Fi(1)p Fh(;k)2931 1535 y Fp(\))p Fo(;)75 1793 y Fp(for)27 b(all)h Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(A)p Fp(\()p Fg(C)862 1760 y Fk(2)902 1793 y Fp(\),)j(and)f(id)f(is)i (the)f(iden)m(tit)m(y)i(op)s(erator.)40 b(As)27 b(a)g(consequence)h(of) f(Lemma)h(2)f(w)m(e)h(get)g(the)75 1906 y(follo)m(wing)k(lemma.)75 2119 y Fc(Lemma)j(5)45 b Ff(Ther)-5 b(e)35 b(exist)g(r)-5 b(e)g(al)36 b(numb)-5 b(ers)35 b Fo(X)r(;)15 b(T)41 b(>)28 b Fp(0)35 b Ff(smal)5 b(l)35 b(enough)g(\(that)h(dep)-5 b(end)36 b(on)e(the)h(c)-5 b(o)g(e\016cients)35 b(of)75 2231 y Fo(a)123 2246 y Fm(k)166 2231 y Fp(\()p Fo(t;)15 b(x)p Fp(\))33 b Ff(for)g Fp(0)26 b Fn(\024)f Fo(k)j Fn(\024)d Fo(S)h Fn(\000)20 b Fp(1)p Ff(\))33 b(such)f(that)i Fn(A)e Ff(is)h(a)g(b)-5 b(ounde)g(d)34 b(line)-5 b(ar)34 b(op)-5 b(er)g(ator,)35 b(fr)-5 b(om)75 2344 y Fp(\()p Fo(G)181 2311 y Fm(q)220 2344 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b Fn(jj)p Fo(:)p Fn(jj)638 2358 y Fm(T)6 b(;X)775 2344 y Fp(\))33 b Ff(into)g(itself.)42 b(Mor)-5 b(e)g(over)33 b(the)g(estimates)h(hold,)1269 2549 y Fn(jjA)p Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)1689 2563 y Fm(T)6 b(;X)1850 2549 y Fn(\024)25 b Fp(\(1)p Fo(=)p Fp(2\))p Fn(jj)p Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)2498 2563 y Fm(T)6 b(;X)2635 2549 y Fo(;)75 2753 y Ff(for)33 b(al)5 b(l)33 b Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)780 2720 y Fm(q)819 2753 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)75 2965 y Fc(Corollary)35 b(1)46 b Ff(Ther)-5 b(e)49 b(exist)g(r)-5 b(e)g(al)50 b(numb)-5 b(ers)50 b Fo(X)r(;)15 b(T)68 b(>)54 b Fp(0)49 b Ff(smal)5 b(l)50 b(enough)f(\(that)h(dep)-5 b(end)51 b(on)e(the)g(c)-5 b(o)g(e\016-)75 3078 y(cients)38 b(of)f Fo(a)502 3093 y Fm(k)545 3078 y Fp(\()p Fo(t;)15 b(x)p Fp(\))39 b Ff(for)f Fp(0)d Fn(\024)f Fo(k)j Fn(\024)d Fo(S)29 b Fn(\000)24 b Fp(1)p Ff(\))38 b(such)g(that)h Fn(D)26 b(\016)f Fo(@)2225 3045 y Fl(\000)p Fm(S)2220 3101 y(x)2368 3078 y Ff(is)38 b(a)g(b)-5 b(ounde)g(d)39 b(invertible)e(op)-5 b(er)g(ator)41 b(fr)-5 b(om)75 3191 y Fp(\()p Fo(G)181 3158 y Fm(q)220 3191 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b Fn(jj)p Fo(:)p Fn(jj)638 3205 y Fm(T)6 b(;X)775 3191 y Fp(\))33 b Ff(into)g(itself.)42 b(In)32 b(p)-5 b(articular,)35 b(ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fo(C)e(>)25 b Fp(0)33 b Ff(such)f(that)1145 3395 y Fn(jj)p Fp(\()p Fn(D)23 b(\016)e Fo(@)1442 3358 y Fl(\000)p Fm(S)1437 3418 y(x)1548 3395 y Fp(\))1583 3358 y Fl(\000)p Fk(1)1678 3395 y Fo(b)p Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)1962 3409 y Fm(T)6 b(;X)2122 3395 y Fn(\024)25 b Fo(C)7 b Fn(jj)p Fo(b)p Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(jj)2624 3409 y Fm(T)6 b(;X)2760 3395 y Fo(;)75 3600 y Ff(for)33 b(al)5 b(l)33 b Fo(b)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)767 3567 y Fm(q)806 3600 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)75 3812 y Fp(The)30 b(main)g(result)g(of)h(this)f(section)i(is)e(the)h (follo)m(wing.)75 4025 y Fc(Theorem)k(1)46 b Ff(Consider)33 b(a)g(p)-5 b(artial)35 b Fo(q)s Fn(\000)p Ff(di\013er)-5 b(enc)g(e)33 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)75 4310 y Fp(\(13\))829 b Fo(@)1117 4273 y Fm(S)1112 4333 y(x)1168 4310 y Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)1537 4196 y Fm(S)t Fl(\000)p Fk(1)1540 4224 y Fj(X)1541 4421 y Fm(k)r Fk(=0)1689 4310 y Fo(a)1737 4325 y Fm(k)1780 4310 y Fp(\()p Fo(t;)15 b(x)p Fp(\)\()p Fo(@)2063 4273 y Fm(k)2058 4333 y(x)2107 4310 y Fo(u)p Fp(\)\()p Fo(q)2273 4273 y Fm(m)2335 4285 y Fi(0)p Fh(;k)2427 4310 y Fo(t;)g(xq)2596 4273 y Fl(\000)p Fm(m)2713 4285 y Fi(1)p Fh(;k)2805 4310 y Fp(\))p Fo(;)75 4598 y Ff(wher)-5 b(e)34 b Fo(S)c Fn(\025)25 b Fp(1)p Ff(,)33 b Fo(m)700 4613 y Fk(0)p Fm(;k)797 4598 y Fo(;)15 b(m)917 4613 y Fk(1)p Fm(;k)1040 4598 y Fn(\025)25 b Fp(0)33 b Ff(ar)-5 b(e)34 b(inte)-5 b(gers)33 b(and)g Fo(a)1931 4613 y Fm(k)1974 4598 y Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fg(C)p Fp([)p Fo(t;)15 b(x)p Fp(])33 b Ff(for)g Fp(0)26 b Fn(\024)f Fo(k)j Fn(\024)d Fo(S)g Fn(\000)20 b Fp(1)p Ff(.)75 4761 y(We)32 b(make)h(the)g(hyp)-5 b(othesis)35 b(that)f(the)f(assumption)i Fc(\(A\))d Ff(is)g(ful\014le) -5 b(d.)75 4924 y(We)32 b(imp)-5 b(ose)34 b(the)f(initial)g(c)-5 b(onditions)35 b(:)41 b(F)-7 b(or)34 b(al)5 b(l)33 b Fp(0)26 b Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(S)g Fn(\000)20 b Fp(1)p Ff(,)75 5128 y Fp(\(14\))1357 b(\()p Fo(@)1680 5090 y Fm(j)1675 5150 y(x)1720 5128 y Fo(u)p Fp(\)\()p Fo(t;)15 b Fp(0\))27 b(=)e Fo(\036)2172 5142 y Fm(j)2208 5128 y Fp(\()p Fo(t)p Fp(\))p Fo(;)75 5332 y Ff(wher)-5 b(e)34 b Fo(\036)386 5346 y Fm(j)422 5332 y Fp(\()p Fo(t)p Fp(\))26 b Fn(2)f Fo(G)708 5299 y Fm(q)747 5332 y Fp(\()p Fo(T)835 5346 y Fk(0)874 5332 y Fo(;)15 b(X)989 5346 y Fk(0)1030 5332 y Fp(\))p Ff(,)32 b(for)i(given)e Fo(T)1561 5346 y Fk(0)1600 5332 y Fo(;)15 b(X)1715 5346 y Fk(0)1781 5332 y Fo(>)24 b Fp(0)p Ff(.)216 5445 y(Then,)34 b(ther)-5 b(e)34 b(exist)g Fo(X)r(;)15 b(T)46 b Ff(smal)5 b(l)35 b(enough)f(such)f(that)i(the)e(pr)-5 b(oblem)36 b(\(13\),)e(\(14\))h (has)f(a)g(unique)e(solution)75 5558 y Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))33 b Ff(in)g Fo(G)538 5525 y Fm(q)576 5558 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Ff(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fp(9)75 399 y Fc(Pro)s(of)31 b Fp(Ev)m(ery)g(function)f Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(A)p Fp(\()p Fg(C)1505 366 y Fk(2)1545 399 y Fp(\))30 b(can)h(b)s(e)f(written)g(in)g(the)h(form)1379 603 y Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)f Fo(@)1801 565 y Fl(\000)p Fm(S)1796 625 y(x)1907 603 y Fo(v)s Fp(\()p Fo(t;)15 b(x)p Fp(\))22 b(+)e Fo(w)r Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fo(;)75 807 y Fp(where)1530 983 y Fo(w)r Fp(\()p Fo(t;)g(x)p Fp(\))27 b(=)1915 870 y Fm(S)t Fl(\000)p Fk(1)1917 897 y Fj(X)1922 1092 y Fm(j)t Fk(=0)2066 983 y Fo(')2125 997 y Fm(j)2162 983 y Fp(\()p Fo(t)p Fp(\))2275 922 y Fo(x)2327 889 y Fm(j)p 2275 962 89 4 v 2285 1046 a Fo(j)5 b Fp(!)2374 983 y Fo(:)75 1245 y Fp(F)-8 b(rom)35 b(the)g(initial)h(conditions)f(\(14\))h(and)e(Lemma)h(4)g(let)g(us)f (assume)h(that)g Fo(w)r Fp(\()p Fo(t;)15 b(x)p Fp(\))36 b(b)s(elongs)e(to)i Fo(G)3562 1212 y Fm(q)3600 1245 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))75 1358 y(for)34 b Fo(X)r(;)15 b(T)45 b(>)31 b Fp(0)j(small)h(enough.)51 b(Then,)34 b(the)h(function)e Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))35 b(is)f(a)h(solution)f(of)h(\(13\),)h(\(14\))g(if)e(and)f (only)h(if)75 1471 y Fo(v)s Fp(\()p Fo(t;)15 b(x)p Fp(\))32 b(satis\014es)e(the)h(equation)75 1675 y(\(15\))1173 b Fn(D)23 b(\016)d Fo(@)1619 1637 y Fl(\000)p Fm(S)1614 1697 y(x)1725 1675 y Fo(v)s Fp(\()p Fo(t;)15 b(x)p Fp(\))27 b(=)e Fn(\000D)s Fo(w)r Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fo(:)75 1879 y Fp(F)-8 b(rom)29 b(the)g(fact)h(that)f Fo(w)r Fp(\()p Fo(t;)15 b(x)p Fp(\))27 b Fn(2)e Fo(G)1286 1846 y Fm(q)1324 1879 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))30 b(is)f(a)g(p)s(olynomial)g(in)g Fo(x)f Fp(of)h(degree)h(less)f (than)f Fo(S)22 b Fn(\000)17 b Fp(1,)30 b(w)m(e)f(deduce)75 1992 y(from)36 b(the)g(lemma)h(3,4)g(that)g Fn(\000D)s Fo(w)r Fp(\()p Fo(t;)15 b(x)p Fp(\))37 b(also)h(b)s(elongs)e(to)h Fo(G)2264 1959 y Fm(q)2302 1992 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\),)39 b(for)d(all)i Fo(X)r(;)15 b(T)48 b(>)35 b Fp(0)i(small)f(enough.)75 2105 y(F)-8 b(rom)25 b(the)f(corollary)h (1,)h(w)m(e)f(deduce)e(that)i(for)f Fo(X)r(;)15 b(T)37 b Fp(small)25 b(enough,)g(there)f(exists)h(a)g(unique)e(solution)h Fo(v)s Fp(\()p Fo(t;)15 b(x)p Fp(\))75 2218 y(of)31 b(\(15\))g(whic)m (h)g(b)s(elongs)f(to)h Fo(G)1139 2185 y Fm(q)1177 2218 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\).)216 2331 y(It)38 b(remains)f(to)h(sho)m(w)f(that)h Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))38 b(b)s(elongs)f(to)h Fo(G)2029 2298 y Fm(q)2068 2331 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))38 b(for)g Fo(X)r(;)15 b(T)50 b Fp(small)38 b(enough.)62 b(Indeed,)38 b(from)75 2444 y(Lemma)j(2,)j(the)d(function)g Fo(@)1111 2411 y Fl(\000)p Fm(S)1106 2466 y(x)1217 2444 y Fo(v)s Fp(\()p Fo(q)1343 2411 y Fm(S)1394 2444 y Fo(t;)1512 2408 y Fm(x)p 1477 2423 109 4 v 1477 2479 a(q)1511 2461 y Fi(2)p Fh(S)1596 2444 y Fp(\))g(b)s(elong)g(to)h Fo(G)2166 2411 y Fm(q)2205 2444 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))42 b(and)e(from)g(Lemma)h(3,)k(the)c(function)75 2579 y Fo(w)r Fp(\()p Fo(q)221 2546 y Fm(S)273 2579 y Fo(t;)15 b(xq)442 2546 y Fl(\000)p Fk(2)p Fm(S)582 2579 y Fp(\))31 b(b)s(elong)g(to)g Fo(G)1121 2546 y Fm(q)1159 2579 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))32 b(for)e Fo(T)8 b(;)15 b(X)38 b Fp(small)31 b(enough.)40 b(So)30 b(that)h(the)g(function)75 2692 y Fo(\027)6 b Fp(\()p Fo(t;)15 b(x)p Fp(\))39 b(=)g Fo(u)p Fp(\()p Fo(q)601 2659 y Fm(S)652 2692 y Fo(t;)15 b(xq)821 2659 y Fl(\000)p Fk(2)p Fm(S)962 2692 y Fp(\))39 b(=)f Fo(@)1198 2659 y Fl(\000)p Fm(S)1193 2714 y(x)1304 2692 y Fo(v)s Fp(\()p Fo(q)1430 2659 y Fm(S)1482 2692 y Fo(t;)1599 2656 y Fm(x)p 1565 2671 V 1565 2728 a(q)1599 2709 y Fi(2)p Fh(S)1684 2692 y Fp(\))26 b(+)f Fo(w)r Fp(\()p Fo(q)1987 2659 y Fm(S)2038 2692 y Fo(t;)15 b(xq)2207 2659 y Fl(\000)p Fk(2)p Fm(S)2348 2692 y Fp(\))39 b(b)s(elongs)f(to)i Fo(G)2947 2659 y Fm(q)2985 2692 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))40 b(for)e Fo(T)8 b(;)15 b(X)46 b Fp(small)75 2827 y(enough.)g(Finally)-8 b(,)34 b(again)g(b)m(y)e(Lemma)g(3,)i(w)m (e)e(deduce)g(that)h Fo(u)p Fp(\()p Fo(t;)15 b(x)p Fp(\))29 b(=)f Fo(\027)6 b Fp(\()2698 2791 y Fm(t)p 2671 2806 79 4 v 2671 2863 a(q)2705 2844 y Fh(S)2760 2827 y Fo(;)15 b(q)2844 2794 y Fk(2)p Fm(S)2930 2827 y Fo(x)p Fp(\))33 b(b)s(elongs)f(to)h Fo(G)3562 2794 y Fm(q)3600 2827 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))75 2940 y(for)30 b Fo(X)r(;)15 b(T)44 b Fp(small)31 b(enough.)2782 b Fa(2)75 3276 y Fq(4)135 b(T)-11 b(ransseries)42 b(solutions)h(of)f(partial)i Fe(q)t Fd(\000)p Fq(di\013erence)f(di\013eren)l(tial)h(equa-)277 3426 y(tions)75 3629 y Fp(W)-8 b(e)32 b(consider)e(a)h(partial)g Fo(q)s Fn(\000)p Fp(di\013erence)f(di\013eren)m(tial)i(equation)75 3914 y(\(16\))837 b Fo(@)1125 3877 y Fm(S)1120 3937 y(z)1176 3914 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))27 b(=)1570 3801 y Fm(S)t Fl(\000)p Fk(1)1573 3828 y Fj(X)1574 4025 y Fm(k)r Fk(=0)1722 3914 y Fo(b)1761 3929 y Fm(k)1804 3914 y Fp(\()p Fo(t;)15 b(z)t Fp(\)\()p Fo(@)2081 3877 y Fm(k)2076 3937 y(z)2125 3914 y Fo(X)7 b Fp(\)\()p Fo(q)2321 3877 y Fm(p)2357 3889 y Fi(0)p Fh(;k)2450 3914 y Fo(t;)15 b(z)t(q)2613 3877 y Fl(\000)p Fm(m)2730 3889 y Fi(1)p Fh(;k)2822 3914 y Fp(\))75 4202 y(where)28 b Fo(q)j Fp(is)d(a)h (complex)g(n)m(um)m(b)s(er)e(suc)m(h)h(that)h Fn(j)p Fo(q)s Fn(j)c Fo(>)g Fp(1)k(and)e Fo(p)2161 4216 y Fk(0)p Fm(;\024)2261 4202 y Fp(,)i Fo(m)2395 4216 y Fk(1)p Fm(;\024)2494 4202 y Fp(,)g Fo(S)k Fp(are)c(in)m(tegers)h(larger)f(than)f(1.)40 b(The)75 4315 y(functions)30 b Fo(b)506 4330 y Fm(k)549 4315 y Fp(\()p Fo(t;)15 b(z)t Fp(\),)31 b(0)26 b Fn(\024)f Fo(k)j Fn(\024)d Fo(S)h Fn(\000)20 b Fp(1)30 b(are)h(of)g(the)f(form) 814 4533 y Fo(b)853 4548 y Fm(k)896 4533 y Fp(\()p Fo(t;)15 b(z)t Fp(\))26 b(=)1229 4447 y Fj(X)1207 4644 y Fm(l)q(;\014)s Fl(\025)p Fk(0)1397 4533 y Fo(b)1436 4548 y Fm(l)q(;\014)s(;k)1583 4533 y Fp(\()p Fo(t)p Fp(\)\()p Fo(e)1763 4547 y Fm(q)1802 4533 y Fp(\()p Fo(t)p Fp(\)\))1940 4495 y Fm(l)1967 4533 y Fo(z)2013 4495 y Fm(\014)2086 4533 y Fp(=)2244 4447 y Fj(X)2182 4648 y Fk(\()p Fm(l)q(;\014)s Fk(\))p Fl(2)p Fm(I)2399 4660 y Fh(k)2452 4533 y Fo(b)2491 4548 y Fm(l)q(;\014)s(;k) 2638 4533 y Fp(\()p Fo(t)p Fp(\)\()p Fo(e)2818 4547 y Fm(q)2857 4533 y Fp(\()p Fo(t)p Fp(\)\))2995 4495 y Fm(l)3022 4533 y Fo(z)3068 4495 y Fm(\014)75 4848 y Fp(where)e Fo(b)371 4863 y Fm(l)q(;\014)s(;k)518 4848 y Fp(\()p Fo(t)p Fp(\))h(are)g(b)s(ounded)d(holomophic)k(functions)e(on)g(the)h (op)s(en)f(set)h Fn(S)7 b Fp(,)26 b(con)m(tin)m(uous)f(on)3348 4825 y(\026)3327 4848 y Fn(S)7 b Fp(,)26 b(considered)75 4961 y(in)33 b(the)h(section)h(2,)g(suc)m(h)e(that)h Fo(b)1203 4976 y Fm(l)q(;\014)s(;k)1350 4961 y Fp(\()p Fo(t)p Fp(\))d(=)f(0,)35 b(for)e(all)i Fo(t)30 b Fn(2)2137 4938 y Fp(\026)2116 4961 y Fn(S)7 b Fp(,)35 b(for)e(\()p Fo(l)r(;)15 b(\014)5 b Fp(\))42 b Fo(=)-55 b Fn(2)30 b Fo(I)2738 4976 y Fm(k)2781 4961 y Fp(,)k(where)f Fo(I)3146 4976 y Fm(k)3189 4961 y Fp(,)h(0)d Fn(\024)g Fo(k)i Fn(\024)d Fo(S)e Fn(\000)22 b Fp(1,)75 5074 y(are)31 b(b)s(ounded)d(subsets)i(of) g Fg(N)1082 5041 y Fk(2)1121 5074 y Fp(.)216 5236 y(W)-8 b(e)33 b(are)f(lo)s(oking)g(for)f(formal)h Ff(tr)-5 b(ansseries)41 b Fp(solutions)32 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))33 b(in)e(the)h(p)s(o)m(w)m(ers)f(of)h Fo(e)3134 5250 y Fm(q)3172 5236 y Fp(\()p Fo(t)p Fp(\))g(and)f Fo(z)t Fp(,)h(of)g(\(16\))75 5349 y(of)f(the)f(form)75 5590 y(\(17\))1107 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))27 b(=)1758 5503 y Fj(X)1736 5701 y Fm(l)q(;\014)s Fl(\025)p Fk(0)1926 5590 y Fo(X)2001 5605 y Fm(l)q(;\014)2090 5590 y Fp(\()p Fo(t)p Fp(\)\()p Fo(e)2270 5604 y Fm(q)2309 5590 y Fp(\()p Fo(t)p Fp(\)\))2447 5552 y Fm(l)2484 5528 y Fo(z)2530 5495 y Fm(\014)p 2484 5569 94 4 v 2490 5652 a Fo(\014)5 b Fp(!)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Fp(10)75 399 y(where)30 b Fo(X)413 414 y Fm(l)q(;\014)502 399 y Fp(\()p Fo(t)p Fp(\))h(are)f(holomorphic)h(functions)f(on)g(the)h(op)s(en)e(set)i Fn(S)7 b Fp(.)216 552 y(By)35 b(inserting)g(the)f(formal)h(transseries) g(\(17\))h(in)e(\(16\),)k(iden)m(tifying)d(the)g(p)s(o)m(w)m(ers)f(of)h Fo(e)3244 566 y Fm(q)3282 552 y Fp(\()p Fo(t)p Fp(\))g(and)f Fo(z)t Fp(,)i(and)75 665 y(using)30 b(the)g(relation)1282 870 y Fo(e)1324 884 y Fm(q)1363 870 y Fp(\()p Fo(q)1442 832 y Fm(p)1478 844 y Fi(0)p Fh(;k)1569 870 y Fo(t)p Fp(\))c(=)1759 743 y Fm(p)1795 755 y Fi(0)p Fh(;k)1882 743 y Fl(\000)p Fk(1)1807 783 y Fj(Y)1800 981 y Fm(h)p Fk(=0)1972 870 y Fp(\(1)21 b(+)f Fo(u)p Fp(\()2299 808 y(1)p 2261 849 122 4 v 2261 932 a Fo(q)2305 906 y Fm(h)2350 932 y Fo(t)2392 870 y Fp(\)\))p Fo(e)2504 884 y Fm(q)2544 870 y Fp(\()p Fo(t)p Fp(\))75 1107 y(for)30 b(the)h(function)f Fo(e)769 1121 y Fm(q)807 1107 y Fp(\()p Fo(t)p Fp(\))h(deduce)f(from)g (\(3\),)i(w)m(e)e(get)i(the)f(follo)m(wing)g(lemma.)75 1292 y Fc(Lemma)k(6)45 b Ff(F)-7 b(or)38 b(given)d(holomorphic)k (functions)e(on)f Fn(S)7 b Ff(,)37 b Fo(X)2211 1307 y Fm(l)q(;j)2289 1292 y Fp(\()p Fo(t)p Fp(\))p Ff(,)h Fo(l)33 b Fn(\025)e Fp(0)p Ff(,)38 b Fp(0)32 b Fn(\024)f Fo(j)37 b Fn(\024)31 b Fo(S)d Fn(\000)22 b Fp(1)p Ff(,)38 b(c)-5 b(onsider)37 b(the)75 1405 y(functions)c Fo(X)545 1420 y Fm(l)q(;\014)634 1405 y Fp(\()p Fo(t)p Fp(\))p Ff(,)g Fo(l)27 b Fn(\025)e Fp(0)p Ff(,)33 b Fo(\014)d Fn(\025)25 b Fo(S)38 b Ff(de\014ne)-5 b(d)33 b(by)g(the)g(fol)5 b(lowing)34 b(r)-5 b(e)g(cursion)33 b(r)-5 b(elation)75 1684 y Fp(\(18\))371 1620 y Fo(X)446 1635 y Fm(l)q(;\014)s Fk(+)p Fm(S)636 1620 y Fp(\()p Fo(t)p Fp(\))p 371 1663 369 4 v 515 1746 a Fo(\014)5 b Fp(!)775 1684 y(=)871 1570 y Fm(S)t Fl(\000)p Fk(1)874 1597 y Fj(X)875 1795 y Fm(k)r Fk(=0)1079 1597 y Fj(X)1023 1795 y Fm(l)1044 1804 y Fi(1)1079 1795 y Fk(+)p Fm(l)1155 1804 y Fi(2)1189 1795 y Fk(=)p Fm(l)1366 1597 y Fj(X)1281 1795 y Fm(\014)1321 1804 y Fi(1)1355 1795 y Fk(+)p Fm(\014)1450 1804 y Fi(2)1484 1795 y Fk(=)p Fm(\014)1607 1620 y Fo(b)1646 1635 y Fm(l)1667 1644 y Fi(1)1702 1635 y Fm(;\014)1762 1644 y Fi(1)1796 1635 y Fm(;k)1858 1620 y Fp(\()p Fo(t)p Fp(\))p Fo(X)2036 1635 y Fm(l)2057 1644 y Fi(2)2092 1635 y Fm(;\014)2152 1644 y Fi(2)2186 1635 y Fk(+)p Fm(k)2284 1620 y Fp(\()p Fo(q)2363 1587 y Fm(p)2399 1599 y Fi(0)p Fh(;k)2490 1620 y Fo(t)p Fp(\))p 1607 1663 952 4 v 2025 1746 a Fo(\014)2076 1760 y Fk(2)2116 1746 y Fp(!)2568 1684 y(\()2603 1557 y Fm(p)2639 1569 y Fi(0)p Fh(;k)2727 1557 y Fl(\000)p Fk(1)2652 1597 y Fj(Y)2645 1795 y Fm(h)p Fk(=0)2817 1684 y Fp(\(1)21 b(+)f Fo(u)p Fp(\()3144 1622 y(1)p 3106 1663 122 4 v 3106 1746 a Fo(q)3150 1720 y Fm(h)3195 1746 y Fo(t)3238 1684 y Fp(\)\)\))3343 1646 y Fm(l)3364 1655 y Fi(2)3403 1684 y Fo(q)3447 1646 y Fl(\000)p Fm(m)3564 1658 y Fi(1)p Fh(;k)3652 1646 y Fm(\014)3692 1655 y Fi(2)75 1957 y Ff(for)33 b(al)5 b(l)33 b Fo(t)25 b Fn(2)g(S)7 b Ff(,)33 b(for)g(al)5 b(l)33 b Fo(\014)d Fn(\025)25 b Fp(0)p Ff(.)42 b(Then)33 b(the)g(formal)i(tr)-5 b(ansseries)34 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))34 b Ff(in)f(\(17\))g(is)g(a)g (solution)h(of)f(\(16\).)216 2142 y Fp(In)h(the)h(next)g(result)f(w)m (e)i(giv)m(e)g(su\016cien)m(t)f(conditions)g(for)f(whic)m(h)h(there)g (exist)g(holomorphic)g(solutions)75 2255 y Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))32 b(of)e(\(16\))i(on)e(the)h(pro)s(duct)e(\()p Fn(S)e(n)20 b Fp(\002\))g Fn(\002)g Fg(C)p Fp(.)40 b(W)-8 b(e)32 b(also)f(study)e(the)i(rate)g(of)f(gro)m(wth)h(of)f(these)h (solutions)75 2368 y(near)f(the)h(singular)f(set)h(\002)20 b Fn(\002)g Fg(C)p Fp(.)75 2522 y(F)-8 b(rom)31 b(Lemma)f(1)h(a\),)h (to)f(eac)m(h)g(in)m(teger)h Fo(p)1513 2537 y Fk(0)p Fm(;k)1635 2522 y Fn(\025)25 b Fp(1)31 b(one)g(can)g(attac)m(h)h(an)e (in)m(teger)i Fo(m)2928 2537 y Fk(0)p Fm(;k)3050 2522 y Fn(\025)25 b Fp(1)31 b(suc)m(h)f(that)1309 2808 y(sup)1317 2896 y Fm(t)p Fl(2)1406 2879 y Fk(\026)1389 2896 y Fl(S)1461 2621 y Fj(\014)1461 2676 y(\014)1461 2731 y(\014)1461 2785 y(\014)1461 2840 y(\014)1461 2894 y(\014)1491 2681 y Fm(p)1527 2693 y Fi(0)p Fh(;k)1614 2681 y Fl(\000)p Fk(1)1540 2721 y Fj(Y)1532 2919 y Fm(h)p Fk(=0)1704 2808 y Fp(\(1)21 b(+)f Fo(u)p Fp(\()2032 2746 y(1)p 1993 2787 V 1993 2870 a Fo(q)2037 2844 y Fm(h)2082 2870 y Fo(t)2125 2808 y Fp(\)\))2195 2621 y Fj(\014)2195 2676 y(\014)2195 2731 y(\014)2195 2785 y(\014)2195 2840 y(\014)2195 2894 y(\014)2251 2808 y Fn(\024)25 b(j)p Fo(q)s Fn(j)2441 2770 y Fm(m)2503 2782 y Fi(0)p Fh(;k)2596 2808 y Fo(:)75 3097 y Fc(Theorem)35 b(2)46 b Ff(Consider)25 b(the)g(p)-5 b(artial)27 b Fo(q)s Fn(\000)p Ff(di\013er)-5 b(enc)g(e)24 b(di\013er)-5 b(ential)26 b(e)-5 b(quation)25 b(\(16\).)41 b(We)24 b(make)h(the)f(fol)5 b(lowing)75 3210 y(assumption.)75 3364 y Fc(Assumption)36 b(\(B\))p Ff(:)75 3477 y(i\))d(The)f(set)h Fn(P)563 3492 y Fm(S)t Fl(\000)p Fm(k)r(;m)786 3504 y Fi(0)p Fh(;k)873 3492 y Fm(;m)955 3504 y Fi(1)p Fh(;k)1047 3477 y Ff(,)f(de\014ne)-5 b(d)34 b(in)e(se)-5 b(ction)33 b(3,)g(is)g(not)g(empty,)75 3590 y(ii\))f Fo(I)240 3605 y Fm(k)308 3590 y Fp(=)25 b Fn(P)467 3605 y Fm(S)t Fl(\000)p Fm(k)r(;m)690 3617 y Fi(0)p Fh(;k)777 3605 y Fm(;m)859 3617 y Fi(1)p Fh(;k)951 3590 y Ff(,)75 3743 y(for)33 b(al)5 b(l)33 b Fp(0)26 b Fn(\024)f Fo(k)j Fn(\024)d Fo(S)h Fn(\000)19 b Fp(1)p Ff(.)75 3897 y(L)-5 b(et)39 b Fo(X)313 3912 y Fm(l)q(;j)391 3897 y Fp(\()p Fo(t)p Fp(\))p Ff(,)h Fo(l)d Fn(\025)e Fp(0)p Ff(,)40 b Fp(0)c Fn(\024)f Fo(j)41 b Fn(\024)35 b Fo(S)29 b Fn(\000)24 b Fp(1)p Ff(,)40 b(b)-5 b(e)38 b(b)-5 b(ounde)g(d)40 b(holomorphic)h(functions)e(on)g Fn(S)7 b Ff(,)39 b(c)-5 b(ontinuous)39 b(on)3786 3874 y Fp(\026)3765 3897 y Fn(S)7 b Ff(,)75 4010 y(satisfying)33 b(the)g(fol)5 b(lowing)34 b(estimates.)43 b(Ther)-5 b(e)33 b(exist)g Fo(C)q(;)15 b(K)33 b(>)25 b Fp(0)33 b Ff(such)f(that)75 4186 y Fp(\(19\))1244 b(sup)1487 4274 y Fm(t)p Fl(2)1576 4257 y Fk(\026)1559 4274 y Fl(S)1631 4186 y Fn(j)p Fo(X)1731 4201 y Fm(l)q(;j)1810 4186 y Fp(\()p Fo(t)p Fp(\))p Fn(j)26 b(\024)f Fo(C)7 b(K)2216 4149 y Fm(l)2241 4186 y Fn(j)p Fo(q)s Fn(j)2335 4149 y Fl(\000)p Fm(l)2412 4125 y Fi(2)75 4423 y Ff(for)33 b(al)5 b(l)33 b Fo(l)27 b Fn(\025)e Fp(0)p Ff(,)33 b(for)g(al)5 b(l)34 b Fp(0)25 b Fn(\024)g Fo(j)31 b Fn(\024)25 b Fo(S)g Fn(\000)20 b Fp(1)p Ff(.)216 4576 y(Then,)47 b(the)d(formal)h(tr)-5 b(ansseries)46 b Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))47 b(=)1858 4508 y Fj(P)1954 4603 y Fm(l)q(;\014)s Fl(\025)p Fk(0)2147 4576 y Fo(X)2222 4591 y Fm(l)q(;\014)2311 4576 y Fp(\()p Fo(t)p Fp(\)\()p Fo(e)2491 4590 y Fm(q)2531 4576 y Fp(\()p Fo(t)p Fp(\)\))2669 4543 y Fm(l)2706 4541 y(z)2742 4517 y Fh(\014)p 2706 4556 78 4 v 2713 4608 a Fm(\014)s Fk(!)2836 4576 y Ff(c)-5 b(onstructe)g(d)46 b(in)d(L)-5 b(emma)45 b(6,)75 4689 y(solution)34 b(of)f(\(16\),)g (de\014nes)g(a)g(holomorphic)j(function)d(on)g(the)g(op)-5 b(en)34 b(set)f Fn(M)25 b Fp(=)g(\()p Fn(S)i(n)21 b Fp(\002\))f Fn(\002)g Fg(C)p Ff(.)216 4843 y(Mor)-5 b(e)g(over,)39 b(the)e(function)f Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))39 b Ff(has)e(the)g(fol)5 b(lowing)38 b(r)-5 b(ate)37 b(of)g(gr)-5 b(owth)39 b(ne)-5 b(ar)37 b(the)g(set)g Fp(\002)23 b Fn(\002)f Fg(C)p Ff(.)54 b(Ther)-5 b(e)75 4956 y(exist)31 b(c)-5 b(onstants)33 b Fo(C)f(>)25 b Fp(0)p Fo(;)15 b(X)34 b(>)25 b Fp(0)31 b Ff(such)g(that)i(for)e(al)5 b(l)32 b Fp(1)26 b Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(s)p Ff(,)30 b Fo(n)25 b Fn(\025)g Fp(0)p Ff(,)32 b(ther)-5 b(e)32 b(exists)f(a)h(c)-5 b(onstant)33 b Fo(C)3593 4970 y Fm(j;n)3713 4956 y Fo(>)25 b Fp(0)75 5069 y Ff(such)33 b(that)75 5261 y Fp(\(20\))517 b(sup)738 5343 y Fl(j)p Fm(z)s Fl(j)p Fm()e Fp(0)p Ff(.)67 b(This)41 b(implies)h(in)75 5627 y(p)-5 b(articular)34 b(that)g Fo(X)7 b Fp(\()p Fo(t;)15 b(z)t Fp(\))33 b Ff(has)g(at)g(most)g(sub-exp)-5 b(onential)33 b(gr)-5 b(owth)35 b(ne)-5 b(ar)33 b(the)f(singular)h (sets)f Fn(f)p Fo(q)3368 5594 y Fl(\000)p Fm(n)3470 5627 y Fo(=n)3570 5641 y Fm(j)3607 5627 y Fn(g)19 b(\002)g Fg(C)p Ff(,)75 5740 y Fp(1)26 b Fn(\024)f Fo(j)31 b Fn(\024)25 b Fo(s)p Ff(,)32 b Fo(n)24 b Fn(\025)h Fp(0)p Ff(.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Fp(11)75 399 y Fc(Pro)s(of)31 b Fp(F)-8 b(rom)31 b(Lemma)g(1,)g(w)m(e)f(get)i(a)f(constan)m(t)g Fo(m)1831 414 y Fk(0)p Fm(;k)1954 399 y Fo(>)25 b Fp(0)31 b(suc)m(h)f(that)1309 686 y(sup)1317 773 y Fm(t)p Fl(2)1406 757 y Fk(\026)1389 773 y Fl(S)1461 499 y Fj(\014)1461 554 y(\014)1461 608 y(\014)1461 663 y(\014)1461 718 y(\014)1461 772 y(\014)1491 559 y Fm(p)1527 571 y Fi(0)p Fh(;k)1614 559 y Fl(\000)p Fk(1)1540 599 y Fj(Y)1532 797 y Fm(h)p Fk(=0)1704 686 y Fp(\(1)21 b(+)f Fo(u)p Fp(\()2032 624 y(1)p 1993 665 122 4 v 1993 748 a Fo(q)2037 722 y Fm(h)2082 748 y Fo(t)2125 686 y Fp(\)\))2195 499 y Fj(\014)2195 554 y(\014)2195 608 y(\014)2195 663 y(\014)2195 718 y(\014)2195 772 y(\014)2251 686 y Fn(\024)25 b(j)p Fo(q)s Fn(j)2441 648 y Fm(m)2503 660 y Fi(0)p Fh(;k)2596 686 y Fo(:)75 974 y Fp(Let)37 b Fo(X)319 989 y Fm(l)q(;j)398 974 y Fp(\()p Fo(t)p Fp(\),)i Fo(l)e Fn(\025)f Fp(0,)j(0)d Fn(\024)g Fo(j)41 b Fn(\024)36 b Fo(S)30 b Fn(\000)24 b Fp(1,)39 b(b)s(e)d(b)s(ounded)e(holomorphic)j(functions)g(on)f Fn(S)7 b Fp(,)39 b(con)m(tin)m(uous)e(on)3813 951 y(\026)3793 974 y Fn(S)75 1087 y Fp(satisfying)k(the)g(estimates)h(\(19\))g(and)e (let)i Fo(X)1673 1102 y Fm(l)q(;\014)1761 1087 y Fp(\()p Fo(t)p Fp(\),)i Fo(l)r(;)15 b(\014)48 b Fn(\025)42 b Fp(0)f(b)s(e)f(the)g(functions)g(constructed)h(from)f(the)75 1200 y(recursion)30 b(relation)h(\(18\).)43 b(In)29 b(the)i(follo)m (wing,)h(w)m(e)f(put)1074 1378 y Fo(w)1139 1393 y Fm(l)q(;\014)1253 1378 y Fp(=)25 b(sup)1358 1466 y Fm(t)p Fl(2)1446 1449 y Fk(\026)1430 1466 y Fl(S)1501 1378 y Fn(j)p Fo(X)1601 1393 y Fm(l)q(;\014)1690 1378 y Fp(\()p Fo(t)p Fp(\))p Fn(j)62 b Fo(;)76 b Fn(j)p Fo(b)p Fn(j)2070 1393 y Fm(l)q(;\014)s(;k) 2242 1378 y Fp(=)25 b(sup)2347 1466 y Fm(t)p Fl(2)2435 1449 y Fk(\026)2419 1466 y Fl(S)2490 1378 y Fn(j)p Fo(b)2554 1393 y Fm(l)q(;\014)s(;k)2701 1378 y Fp(\()p Fo(t)p Fp(\))p Fn(j)p Fo(;)75 1620 y Fp(for)33 b(all)h Fo(l)r(;)15 b(\014)35 b Fn(\025)30 b Fp(0)k(and)e(0)e Fn(\024)g Fo(k)j Fn(\024)d Fo(S)d Fn(\000)22 b Fp(1.)49 b(By)34 b(construction,)g(w)m(e)g(ha)m(v)m (e)g(that)g Fn(j)p Fo(b)p Fn(j)2845 1635 y Fm(l)q(;\014)s(;k)3022 1620 y Fp(=)c(0)j(if)g(\()p Fo(l)r(;)15 b(\014)5 b Fp(\))42 b Fo(=)-55 b Fn(2)29 b Fo(I)3644 1635 y Fm(k)3687 1620 y Fp(,)34 b(for)75 1733 y(0)26 b Fn(\024)f Fo(k)j Fn(\024)d Fo(S)g Fn(\000)20 b Fp(1.)75 1887 y(F)-8 b(rom)38 b(the)f(fact)h(that)g (the)f(set)h Fn(S)43 b Fp(is)38 b(in)m(v)-5 b(arian)m(t)38 b(under)d(the)j(dilations)g Fo(t)e Fn(7!)g Fo(q)2815 1854 y Fm(p)2851 1866 y Fi(0)p Fh(;k)2942 1887 y Fo(t)p Fp(,)j(w)m(e)e(get)i(the)e(follo)m(wing)75 2000 y(inequalities)32 b(for)e(the)g(n)m(um)m(b)s(ers)f Fo(w)1280 2015 y Fm(l)q(;\014)1369 2000 y Fp(,)75 2269 y(\(21\))756 2206 y Fo(w)821 2221 y Fm(l)q(;\014)s Fk(+)p Fm(S)p 756 2248 256 4 v 843 2331 a Fo(\014)5 b Fp(!)1046 2269 y Fn(\024)1142 2155 y Fm(S)t Fl(\000)p Fk(1)1145 2183 y Fj(X)1146 2380 y Fm(k)r Fk(=0)1350 2183 y Fj(X)1294 2380 y Fm(l)1315 2389 y Fi(1)1350 2380 y Fk(+)p Fm(l)1426 2389 y Fi(2)1461 2380 y Fk(=)p Fm(l)1637 2183 y Fj(X)1552 2380 y Fm(\014)1592 2389 y Fi(1)1627 2380 y Fk(+)p Fm(\014)1722 2389 y Fi(2)1756 2380 y Fk(=)p Fm(\014)1879 2206 y Fn(j)p Fo(b)p Fn(j)1968 2221 y Fm(l)1989 2230 y Fi(1)2024 2221 y Fm(;\014)2084 2230 y Fi(1)2118 2221 y Fm(;k)2180 2206 y Fo(w)2245 2221 y Fm(l)2266 2230 y Fi(2)2301 2221 y Fm(;\014)2361 2230 y Fi(2)2394 2221 y Fk(+)p Fm(k)p 1879 2248 614 4 v 2127 2331 a Fo(\014)2178 2345 y Fk(2)2218 2331 y Fp(!)2502 2269 y Fn(j)p Fo(q)s Fn(j)2596 2231 y Fm(m)2658 2243 y Fi(0)p Fh(;k)2746 2231 y Fm(l)2767 2240 y Fi(2)2806 2269 y Fn(j)p Fo(q)s Fn(j)2900 2231 y Fl(\000)p Fm(m)3017 2243 y Fi(1)p Fh(;k)3105 2231 y Fm(\014)3145 2240 y Fi(2)75 2538 y Fp(for)30 b(all)h Fo(l)r(;)15 b(\014)31 b Fn(\025)25 b Fp(0.)216 2651 y(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(put)643 2838 y Fo(\036)697 2852 y Fm(j)733 2838 y Fp(\()p Fo(t)p Fp(\))26 b(=)958 2751 y Fj(X)968 2949 y Fm(l)q Fl(\025)p Fk(0)1104 2838 y Fo(w)1169 2853 y Fm(l)q(;j)1248 2838 y Fo(t)1281 2800 y Fm(l)1367 2838 y Fo(;)76 b Fn(j)p Fo(b)p Fn(j)1557 2853 y Fm(k)1600 2838 y Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b(=)1939 2751 y Fj(X)1917 2949 y Fm(l)q(;\014)s Fl(\025)p Fk(0)2107 2838 y Fn(j)p Fo(b)p Fn(j)2196 2853 y Fm(l)q(;\014)s(;k)2343 2838 y Fo(t)2376 2800 y Fm(l)2402 2838 y Fo(x)2454 2800 y Fm(\014)2527 2838 y Fp(=)2684 2751 y Fj(X)2622 2953 y Fk(\()p Fm(l)q(;\014)s Fk(\))p Fl(2)p Fm(I)2839 2965 y Fh(k)2893 2838 y Fn(j)p Fo(b)p Fn(j)2982 2853 y Fm(l)q(;\014)s(;k) 3129 2838 y Fo(t)3162 2800 y Fm(l)3188 2838 y Fo(x)3240 2800 y Fm(\014)75 3115 y Fp(for)k(all)h(0)26 b Fn(\024)f Fo(j;)15 b(k)30 b 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Fp(1)p Fo(:)75 4140 y Fp(F)-8 b(rom)22 b(the)g(estimates)i(\(19\),)h(w)m(e)e(get)g(in)e(particular)i(that)f (the)g(series)g Fo(\036)2458 4154 y Fm(j)2495 4140 y Fp(\()p Fo(t)p Fp(\))g(b)s(elong)g(to)h(the)f(space)g Fo(G)3458 4107 y Fl(j)p Fm(q)r Fl(j)3536 4140 y Fp(\()p Fo(T)3624 4154 y Fk(0)3664 4140 y Fo(;)15 b(X)3779 4154 y Fk(0)3819 4140 y Fp(\))75 4252 y(for)34 b Fo(T)271 4266 y Fk(0)343 4252 y Fo(>)d Fp(0)k(small)g(enough)f(and)g Fo(X)1340 4266 y Fk(0)1411 4252 y Fo(>)e Fp(0.)53 b(F)-8 b(rom)35 b(the)g(assumption)e Fc(\(B\))p Fp(,)i(the)f(set)h Fn(P)3110 4267 y Fm(S)t Fl(\000)p Fm(k)r(;m)3333 4279 y Fi(0)p Fh(;k)3420 4267 y Fm(;m)3502 4279 y Fi(1)p Fh(;k)3628 4252 y Fp(is)f(not)75 4365 y(empt)m(y)c(and)e(the)i(p)s(olynomial)f Fn(j)p Fo(b)p Fn(j)1240 4380 y Fm(k)1283 4365 y Fp(\()p Fo(t;)15 b(x)p Fp(\))30 b(is)g(of)f Fn(P)1764 4380 y Fm(S)t Fl(\000)p Fm(k)r(;m)1987 4392 y Fi(0)p Fh(;k)2074 4380 y Fm(;m)2156 4392 y Fi(1)p Fh(;k)2247 4365 y Fn(\000)p Fp(t)m(yp)s(e,)h(for)f(all)h(0)c 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Fi(0)p Fh(;k)2713 5615 y Fm(l)2734 5624 y Fi(2)2772 5653 y Fn(j)p Fo(q)s Fn(j)2866 5615 y Fl(\000)p Fm(m)2983 5627 y Fi(1)p Fh(;k)3072 5615 y Fm(\014)3112 5624 y Fi(2)3150 5653 y Fo(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Fp(12)75 399 y(for)30 b(all)h Fo(l)r(;)15 b(\014)31 b Fn(\025)25 b Fp(0,)31 b(with)f(initial)i(data)75 582 y(\(25\))991 b Fo(v)1270 597 y Fm(l)q(;j)1373 582 y Fp(=)25 b Fo(w)1534 597 y Fm(l)q(;j)1672 582 y Fo(;)76 b(l)28 b Fn(\025)d Fp(0)61 b Fo(;)76 b Fp(0)25 b Fn(\024)g Fo(j)31 b Fn(\024)25 b Fo(S)g Fn(\000)20 b Fp(1)p Fo(:)75 765 y Fp(F)-8 b(rom)31 b(the)f(relations)i(\(21\),)g(\(24\))g(and)e(\(25\),)i(w)m(e)f(get)g (that)75 948 y(\(26\))1526 b Fo(w)1826 963 y Fm(l)q(;\014)1940 948 y Fn(\024)25 b Fo(v)2080 963 y Fm(l)q(;\014)75 1131 y Fp(for)30 b(all)h Fo(l)r(;)15 b(\014)31 b Fn(\025)25 b Fp(0.)41 b(In)29 b(particular,)i(from)e(the)h(fact)h(that)g Fo(V)20 b Fp(\()p Fo(t;)15 b(x)p Fp(\))26 b Fn(2)f Fo(G)2453 1098 y Fl(j)p Fm(q)r Fl(j)2531 1131 y Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))31 b(for)f Fo(T)8 b(;)15 b(X)33 b(>)25 b Fp(0)30 b(small)h(enough)75 1244 y(and)f(the)g(inequalit)m(y)i Fo(l)r(\014)e Fn(\024)25 b Fo(l)1065 1211 y Fk(2)1105 1244 y Fo(=)p Fp(2)c(+)f Fo(\014)1363 1211 y Fk(2)1402 1244 y Fo(=)p Fp(2,)32 b(w)m(e)f(get)g(that)g(there)g(exist)g Fo(C)q(;)15 b(T)8 b(;)15 b(X)34 b(>)25 b Fp(0)30 b(suc)m(h)g(that)75 1465 y(\(27\))563 b Fn(j)p Fo(w)888 1480 y Fm(l)q(;\014)977 1465 y Fn(j)25 b(\024)g Fo(C)7 b Fn(j)p Fo(q)s Fn(j)1289 1427 y Fm(P)j Fk(\()p Fm(l)q(;\014)s Fk(\))1487 1465 y Fo(\014)5 b Fp(!\()1624 1403 y(1)p 1613 1444 66 4 v 1613 1527 a Fo(T)1690 1465 y Fp(\))1725 1427 y Fm(l)1751 1465 y Fp(\()1815 1403 y(1)p 1796 1444 83 4 v 1796 1527 a Fo(X)1889 1465 y Fp(\))1924 1427 y Fm(\014)1997 1465 y Fn(\024)25 b Fo(C)7 b Fn(j)p Fo(q)s Fn(j)2259 1427 y Fl(\000)p Fm(l)2336 1404 y Fi(2)2370 1427 y Fm(=)p Fk(2)p Fl(\000)p Fm(\014)2538 1404 y Fi(2)2573 1427 y Fm(=)p Fk(2)2647 1465 y Fo(\014)e Fp(!\()2784 1403 y(1)p 2773 1444 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b(P)-8 b(.)28 b(Ramis,)h(C.)f(Zhang,)h Ff(D)n(\023)-44 b(evelopp)-5 b(ement)31 b(asymptotique)h Fo(q)s Fn(\000)p Ff(Gevr)-5 b(ey)31 b(et)f(fonction)i(th)n(^)-44 b(eta)31 b(de)g(Jac)-5 b(obi.)262 3232 y Fp(C.)30 b(R.)h(Math.)g(Acad.)g(Sci.)g(P)m(aris)f (335)i(\(2002\),)h(no.)e(11,)g(899{902.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0712040922555--