(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips 5.511 Copyright 1986, 1993 Radical Eye Software %%Title: All.dvi %%CreationDate: Wed Jul 24 12:29:20 1996 %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -pp 59-65 All.dvi -o /home/blagny/lib/algo/seminars/sem92-93/salvy2.ps %DVIPSSource: TeX output 1994.01.03:1834 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution hsize -72 div mul 0 TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook} if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255 {IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{ pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{ gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{/CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{10 div /rwi X /rwiSeen true N}B /@rhi {10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{ }N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp{pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray}N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{ SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{ rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin}N /@fedspecial{end}B /li{lineto}B /rl{ rlineto}B /rc{rcurveto}B /np{/SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (/a/home/pommard/algo/salvy/Tex/Sem/Sem93/All.dvi) @start /Fa 20 121 df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b 1 82 df<001FE00000F03C00 03601B000440088008800440108004202100021021000210410002084200010842000108820001 048200010482000104820001048200010482000104820001048200010442000108420001084100 0208210002102100021010800420088004400440088003601B0000F03C00001FE0000010400000 08200000042000000310000000CC0000003FF81E247F9C23>81 D E /Fc 3 107 df<0200020002000200C2183FE00F8007000D80088010400D0B7F8A0F>63 D<0808000000007098B0303060646870060F7D8E0B>105 D<00C0008000000000000000000F00 11801180030003000300030006000600060006008C00F0000A137F8E0C>I E /Fd 4 55 df<3E00418080C0C0C000C000C0018003000400084030407F80FF800A0D7E8C0E> 50 D<0300070007000B00130023006300C300FFC00300030003001FC00A0D7E8C0E>52 D<20803F003C00200020003F00218000C000C0C0C080C041803E000A0D7E8C0E>I<0F00118021 806000C000DE00E180C0C0C0C0C0C060C021801E000A0D7E8C0E>I E /Fe 13 126 df<0000300000600000C0000180000300000700000E00000C0000180000380000300000 700000E00000C00001C0000180000380000380000300000700000600000E00000E00000C00001C 00001C00001C000018000038000038000038000038000070000070000070000070000070000070 0000700000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E0000070000070000070000070 00007000007000007000003800003800003800003800001800001C00001C00001C00000C00000E 00000E000006000007000003000003800003800001800001C00000C00000E00000700000300000 3800001800000C00000E000007000003000001800000C0000060000030146377811F>18 DI<00001800003000 00600000E00000C0000180000380000700000600000E00000C00001C0000380000380000700000 700000E00000E00001E00001C00001C0000380000380000380000780000700000700000F00000E 00000E00001E00001E00001E00001C00001C00003C00003C00003C00003C000038000078000078 0000780000780000780000780000780000780000700000F00000F00000F00000F00000F00000F0 0000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0 0000F00000F00000F00000F00000F00000F00000F0000070000078000078000078000078000078 00007800007800007800003800003C00003C00003C00003C00001C00001C00001E00001E00001E 00000E00000E00000F000007000007000007800003800003800003800001C00001C00001E00000 E00000E000007000007000003800003800001C00000C00000E0000060000070000038000018000 00C00000E0000060000030000018157C768121>32 DI80 DI<0000F800018400030600060E00060400 0E00000E00000E00000C00001C00001C00001C00001C00001C00001C00001C00001C00001C0000 380000380000380000380000380000380000380000380000380000380000700000700000700000 700000700000700000700000700000700000600000E00000E00000E00040C000E0C000C1800043 00003E0000172E7E7F14>I88 D<000000038000000006600000000C700000000CF00000000CF00000001C600000001800000000 3800000000380000000038000000007000000000700000000070000000007000000000F0000000 00E000000000E000000000E000000001E000000001E000000001C000000001C000000003C00000 0003C000000003C000000003C000000007800000000780000000078000000007800000000F8000 00000F800000000F000000000F000000001F000000001F000000001F000000001F000000001E00 0000003E000000003E000000003E000000003E000000003C000000007C000000007C000000007C 000000007C000000007800000000F800000000F800000000F800000000F800000000F000000001 F000000001F000000001F000000001F000000001E000000001E000000003E000000003E0000000 03C000000003C000000003C000000003C000000007800000000780000000078000000007800000 00070000000007000000000F000000000F000000000E000000000E000000000E000000001E0000 00001C000000001C000000001C0000000018000000003800000000380000000030000000007000 00006060000000F060000000F0C0000000E18000000063000000001E00000000245C7E7F17>90 D<0001F8000FF8003FF800FFF801FFF803FE0007E0000F80001E0000380000700000600000E000 00C00000150E818413>122 DII<000018000038 0000300000700000E00003C0000F80003F0003FE00FFFC00FFF800FFE000FF8000FC0000150E81 8D13>I E /Ff 6 107 df0 D<040E0E1C1C1C38383070706060C0 C0070F7F8F0A>48 D<03FC0FFC1C003000600060006000C000C000FFFCFFFCC000C00060006000 600030001C000FFC03FC0E147D9016>50 D<00E003000600060006000600060006000600060006 00060006001C00F0001C0006000600060006000600060006000600060006000600030000E00B1D 7E9511>102 DI106 D E /Fg 6 113 df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h 50 123 df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i 16 123 df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j 33 122 df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k 20 122 df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l 13 62 df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m 7 55 df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n 10 122 df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o 17 122 df73 D<07FC001FFF803F07C03F03E03F01E03F01F01E01F00001F00001F0003FF003FDF01FC1F03F01 F07E01F0FC01F0FC01F0FC01F0FC01F07E02F07E0CF81FF87F07E03F18167E951B>97 DI<00FF8007FFE00F83 F01F03F03E03F07E03F07C01E07C0000FC0000FC0000FC0000FC0000FC0000FC00007C00007E00 007E00003E00301F00600FC0E007FF8000FE0014167E9519>I<0001FE000001FE0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E00 01FC3E0007FFBE000F81FE001F007E003E003E007E003E007C003E00FC003E00FC003E00FC003E 00FC003E00FC003E00FC003E00FC003E00FC003E007C003E007C003E003E007E001E00FE000F83 BE0007FF3FC001FC3FC01A237EA21F>I<00FE0007FF800F87C01E01E03E01F07C00F07C00F8FC 00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E00003E00181F00300FC07003 FFC000FF0015167E951A>I<03FC1E0FFF7F1F0F8F3E07CF3C03C07C03E07C03E07C03E07C03E0 7C03E03C03C03E07C01F0F801FFF0013FC003000003000003800003FFF801FFFF00FFFF81FFFFC 3800FC70003EF0001EF0001EF0001EF0001E78003C7C007C3F01F80FFFE001FF0018217E951C> 103 D<1C003E007F007F007F003E001C000000000000000000000000000000FF00FF001F001F00 1F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F00FFE0FFE00B247E A310>105 D108 DII<00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C007CFC007EFC007EFC00 7EFC007EFC007EFC007EFC007E7C007C7C007C3E00F81F01F00F83E007FFC000FE0017167E951C >I114 D<0FF3003FFF00781F00600700E00300E00300F00300FC00007FE0007FF8003FFE000FFF0001FF 00000F80C00780C00380E00380E00380F00700FC0E00EFFC00C7F00011167E9516>I<01800001 80000180000180000380000380000780000780000F80003F8000FFFF00FFFF000F80000F80000F 80000F80000F80000F80000F80000F80000F80000F80000F80000F81800F81800F81800F81800F 81800F830007C30003FE0000F80011207F9F16>II121 D E /Fp 76 127 df<007E1F0001C1B1800303E3C0 0703C3C00E03C1800E01C0000E01C0000E01C0000E01C0000E01C0000E01C000FFFFFC000E01C0 000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01 C0000E01C0000E01C0000E01C0000E01C0000E01C0007F87FC001A1D809C18>11 D<007E0001C1800301800703C00E03C00E01800E00000E00000E00000E00000E0000FFFFC00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C07F87F8151D809C17>I<003F07E00001C09C18000380F018000701F03C000E01 E03C000E00E018000E00E000000E00E000000E00E000000E00E000000E00E00000FFFFFFFC000E 00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C00 0E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C 007FC7FCFF80211D809C23>14 D<6060F0F0F8F86868080808080808101010102020404080800D 0C7F9C15>34 D<60F0F8680808081010204080050C7C9C0C>39 D<004000800100020006000C00 0C0018001800300030007000600060006000E000E000E000E000E000E000E000E000E000E000E0 00E000600060006000700030003000180018000C000C00060002000100008000400A2A7D9E10> I<800040002000100018000C000C000600060003000300038001800180018001C001C001C001C0 01C001C001C001C001C001C001C001C0018001800180038003000300060006000C000C00180010 002000400080000A2A7E9E10>I<00060000000600000006000000060000000600000006000000 060000000600000006000000060000000600000006000000060000FFFFFFE0FFFFFFE000060000 000600000006000000060000000600000006000000060000000600000006000000060000000600 0000060000000600001B1C7E9720>43 D<60F0F0701010101020204080040C7C830C>II<60F0F06004047C830C>I<03C00C301818300C300C700E60066006E007E007E0 07E007E007E007E007E007E007E007E007E007E00760066006700E300C300C18180C3007E0101D 7E9B15>48 D<030007003F00C70007000700070007000700070007000700070007000700070007 000700070007000700070007000700070007000F80FFF80D1C7C9B15>I<07C01830201C400C40 0EF00FF80FF807F8077007000F000E000E001C001C00380070006000C00180030006010C011801 10023FFE7FFEFFFE101C7E9B15>I<07E01830201C201C781E780E781E381E001C001C00180030 006007E00030001C001C000E000F000F700FF80FF80FF80FF00E401C201C183007E0101D7E9B15 >I<000C00000C00001C00003C00003C00005C0000DC00009C00011C00031C00021C00041C000C 1C00081C00101C00301C00201C00401C00C01C00FFFFC0001C00001C00001C00001C00001C0000 1C00001C0001FFC0121C7F9B15>I<300C3FF83FF03FC020002000200020002000200023E02430 2818301C200E000E000F000F000F600FF00FF00FF00F800E401E401C2038187007C0101D7E9B15 >I<00F0030C06040C0E181E301E300C700070006000E3E0E430E818F00CF00EE006E007E007E0 07E007E007600760077006300E300C18180C3003E0101D7E9B15>I<4000007FFF807FFF007FFF 0040020080040080040080080000100000100000200000600000400000C00000C00001C0000180 00018000038000038000038000038000078000078000078000078000078000078000030000111D 7E9B15>I<03E00C301008200C20066006600660067006780C3E083FB01FE007F007F818FC307E 601E600FC007C003C003C003C00360026004300C1C1007E0101D7E9B15>I<03C00C301818300C 700C600EE006E006E007E007E007E007E0076007700F300F18170C2707C700060006000E300C78 0C78187010203030C00F80101D7E9B15>I<60F0F0600000000000000000000060F0F06004127C 910C>I<60F0F0600000000000000000000060F0F0701010101020204080041A7C910C>I<7FFFFF C0FFFFFFE00000000000000000000000000000000000000000000000000000000000000000FFFF FFE07FFFFFC01B0C7E8F20>61 D<000600000006000000060000000F0000000F0000000F000000 17800000178000001780000023C0000023C0000023C0000041E0000041E0000041E0000080F000 0080F0000180F8000100780001FFF80003007C0002003C0002003C0006003E0004001E0004001E 000C001F001E001F00FF80FFF01C1D7F9C1F>65 DI<001F808000 E0618001801980070007800E0003801C0003801C00018038000180780000807800008070000080 F0000000F0000000F0000000F0000000F0000000F0000000F0000000F000000070000080780000 8078000080380000801C0001001C0001000E000200070004000180080000E03000001FC000191E 7E9C1E>IIII<001F808000E0 618001801980070007800E0003801C0003801C00018038000180780000807800008070000080F0 000000F0000000F0000000F0000000F0000000F0000000F000FFF0F0000F807000078078000780 78000780380007801C0007801C0007800E00078007000B800180118000E06080001F80001C1E7E 9C21>III<1FFF00F80078007800 7800780078007800780078007800780078007800780078007800780078007800787078F878F878 F878F0F040E021C01F00101D7F9B15>IIIII<003F800000E0E000038038000700 1C000E000E001C0007003C00078038000380780003C0780003C0700001C0F00001E0F00001E0F0 0001E0F00001E0F00001E0F00001E0F00001E0F00001E0700001C0780003C0780003C038000380 3C0007801C0007000E000E0007001C000380380000E0E000003F80001B1E7E9C20>II82 D<07E0801C1980300580700380600180E00180E00080E000 80E00080F00000F800007C00007FC0003FF8001FFE0007FF0000FF80000F800007C00003C00001 C08001C08001C08001C0C00180C00180E00300D00200CC0C0083F800121E7E9C17>I<7FFFFFC0 700F01C0600F00C0400F0040400F0040C00F0020800F0020800F0020800F0020000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E>II87 D89 D91 D<08081010202040404040808080808080B0B0F8F8787830300D 0C7A9C15>II<1FC000307000783800781C00301C00001C00001C0001FC00 0F1C00381C00701C00601C00E01C40E01C40E01C40603C40304E801F870012127E9115>97 DI<07E00C301878307870306000E000E000E000E000E000 E00060007004300418080C3007C00E127E9112>I<003F00000700000700000700000700000700 00070000070000070000070000070003E7000C1700180F00300700700700600700E00700E00700 E00700E00700E00700E00700600700700700300700180F000C370007C7E0131D7E9C17>I<03E0 0C301818300C700E6006E006FFFEE000E000E000E00060007002300218040C1803E00F127F9112 >I<00F8018C071E061E0E0C0E000E000E000E000E000E00FFE00E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E007FE00F1D809C0D>I<00038003C4C00C38C01C38 80181800381C00381C00381C00381C001818001C38000C300013C0001000003000001800001FF8 001FFF001FFF803003806001C0C000C0C000C0C000C06001803003001C0E0007F800121C7F9215 >II<18003C003C00180000000000000000000000000000 00FC001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF80091D 7F9C0C>I107 DIII<03F0000E1C00180600300300700380600180E001C0E001C0E001C0 E001C0E001C0E001C06001807003803003001806000E1C0003F00012127F9115>II<03C1000C3300180B00300F00700700700700E00700E00700E00700E00700E00700E0070060 0700700700300F00180F000C370007C70000070000070000070000070000070000070000070000 3FE0131A7E9116>II<1F9030704030C010C010E010F8007F803FE00FF000F880388018 C018C018E010D0608FC00D127F9110>I<04000400040004000C000C001C003C00FFE01C001C00 1C001C001C001C001C001C001C001C101C101C101C101C100C100E2003C00C1A7F9910>IIII<7F8FF00F03800F030007020003840001C80001D80000F00000700000780000F800009C 00010E00020E000607000403801E07C0FF0FF81512809116>II<7FFC70386038 407040F040E041C003C0038007000F040E041C043C0C380870087038FFF80E127F9112>II<1C043F0843F080E00E047D9B15>126 D E /Fq 44 124 df<003FC00001F0300003C0380007C07C000F807C000F807C000F8038000F8000000F8000000F 8000000F800000FFFFFC00FFFFFC000F807C000F807C000F807C000F807C000F807C000F807C00 0F807C000F807C000F807C000F807C000F807C000F807C000F807C000F807C007FE1FF807FE1FF 80191D809C1B>12 D<78FCFCFEFE7A0202040408083040070E7D9C0D>39 D45 D<78FCFCFCFC7806067D850D>I<00600001E0000FE000 FFE000F3E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E0007FFF807FFF80111B7D9A18> 49 D<07F8001FFE00383F80780FC0FC07C0FC07E0FC03E0FC03E07803E00007E00007C00007C0 000F80001F00001E0000380000700000E0000180600300600600600800E01FFFC03FFFC07FFFC0 FFFFC0FFFFC0131B7E9A18>I<03F8001FFE003C1F003C0F807C07C07E07C07C07C03807C0000F 80000F80001E00003C0003F800001E00000F800007C00007C00007E03007E07807E0FC07E0FC07 E0FC07C0780F80781F001FFE0007F800131B7E9A18>I<000180000380000780000F80001F8000 3F80006F8000CF80008F80018F80030F80060F800C0F80180F80300F80600F80C00F80FFFFF8FF FFF8000F80000F80000F80000F80000F80000F8001FFF801FFF8151B7F9A18>I<78FCFCFCFC78 00000000000070F8FCFCFC7C0404080808102040061A7D910D>59 D66 D68 DII<000FF008007FFE3801FC07F807E001F80F8000781F0000783F0000383E0000 387E0000187C000018FC000000FC000000FC000000FC000000FC000000FC000000FC007FFFFC00 7FFF7C0001F87E0001F83E0001F83F0001F81F0001F80F8001F807E001F801FC07F8007FFE7800 0FF818201C7D9B26>III77 D<003FE00001F07C0003C01E 000F800F801F0007C01E0003C03E0003E07E0003F07C0001F07C0001F0FC0001F8FC0001F8FC00 01F8FC0001F8FC0001F8FC0001F8FC0001F8FC0001F87C0001F07E0003F07E0003F03E0003E03F 0007E01F0007C00F800F8003C01E0001F07C00003FE0001D1C7D9B24>79 D82 D<07F8201FFEE03C07E07801E07000E0F000E0F00060F00060F80000FE0000FFE0007FFE003FFF 003FFF800FFFC007FFE0007FE00003F00001F00000F0C000F0C000F0C000E0E000E0F001C0FC03 C0EFFF0083FC00141C7D9B1B>I<7FFFFFE07FFFFFE0781F81E0701F80E0601F8060E01F8070C0 1F8030C01F8030C01F8030C01F8030001F8000001F8000001F8000001F8000001F8000001F8000 001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F80 0007FFFE0007FFFE001C1C7E9B21>I<0FF8001C1E003E0F803E07803E07C01C07C00007C0007F C007E7C01F07C03C07C07C07C0F807C0F807C0F807C0780BC03E13F80FE1F815127F9117>97 DI<03FC000E0E001C1F003C1F00781F00780E00F80000F8 0000F80000F80000F80000F800007800007801803C01801C03000E0E0003F80011127E9115>I< 000FF0000FF00001F00001F00001F00001F00001F00001F00001F00001F00001F001F9F00F07F0 1C03F03C01F07801F07801F0F801F0F801F0F801F0F801F0F801F0F801F07801F07801F03C01F0 1C03F00F0FFE03F9FE171D7E9C1B>I<01FC000F07001C03803C01C07801C07801E0F801E0F801 E0FFFFE0F80000F80000F800007800007C00603C00601E00C00F038001FC0013127F9116>I<00 7F0001E38003C7C00787C00F87C00F83800F80000F80000F80000F80000F8000FFF800FFF8000F 80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F 80007FF8007FF800121D809C0F>I<03F8F00E0F381E0F381C07303C07803C07803C07803C0780 1C07001E0F000E0E001BF8001000001800001800001FFF001FFFC00FFFE01FFFF07801F8F00078 F00078F000787000707800F01E03C007FF00151B7F9118>II<1E003F003F003F003F001E00000000000000000000000000FF00FF001F001F001F001F001F 001F001F001F001F001F001F001F001F001F00FFE0FFE00B1E7F9D0E>I107 DIII<01FC 000F07801C01C03C01E07800F07800F0F800F8F800F8F800F8F800F8F800F8F800F87800F07800 F03C01E01E03C00F078001FC0015127F9118>II114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007CC01CC0 1CE01CE018F830CFC00E127E9113>I<0300030003000300070007000F000F003FFCFFFC1F001F 001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E1A7F9913>II119 D121 D<3FFF803C1F00303F00303E00607C0060FC0060F80001F00003F00007E00007C1800F81801F81 801F03803E03007E07007C0F00FFFF0011127F9115>II E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 59 1 59 0 bop 528 284 a Fo(In)n(tro)r(duction)18 b(to)g(sym)n(b)r(olic)e(in)n (tegration)843 406 y Fn(Bruno)g(Salvy)781 470 y Fp(INRIA)d(Ro)q(cquencourt) 835 563 y(Marc)o(h)i(22,)e(1993)723 656 y([summa)o(ry)f(b)o(y)h(Daniel)g (Augot])723 841 y Fq(1.)24 b(General)14 b(In)o(tro)q(ductio)o(n)50 917 y Fp(Scien)o(ti\014c)e(comm)o(unit)o(y)c(is)k(b)q(ecoming)e(a)o(w)o(are)i (of)f(the)h(p)q(o)o(w)o(er)g(and)g(utilit)o(y)e(of)h(sym)o(b)q(olic)f (computations)g(soft)o(w)o(ares.)18 b(One)0 967 y(of)13 b(the)i(most)d (surprising)j(ac)o(hiev)o(emen)o(ts)e(of)g(these)j(programs)c(is)i(the)h (abilit)o(y)d(to)i(p)q(erform)f(formal)e(in)o(tegration,)i(that)h(is:)149 1032 y(Giv)o(en)f(a)g(function)h Fh(f)t Fp(\()p Fh(z)r Fp(\))h(\(p)q(ossibly) f(with)f(parameters\),)h(\014nd)g(a)f(function)h Fh(F)6 b Fp(\()p Fh(z)r Fp(\))14 b(suc)o(h)g(that:)853 1121 y Fh(F)886 1104 y Ff(0)880 1131 y Fi(z)899 1121 y Fp(\()p Fh(z)r Fp(\))e(=)f Fh(f)t Fp(\()p Fh(z)r Fp(\))p Fh(:)-947 b Fp(\(1\))50 1206 y(Most)18 b(mathematical)d(studen)o(ts)20 b(ha)o(v)o(e)e(learned)g(ho)o(w)g (to)g(\014nd)g(suc)o(h)h(primitiv)o(es,)e(generally)h(with)g(heuristics.)32 b(The)0 1255 y(imp)q(ortan)o(t)12 b(p)q(oin)o(t)i(here)h(is)e(that)h(sym)o(b) q(olic)e(soft)o(w)o(ares)j(\(lik)o(e)e(Maple\))h(use)g Fj(algorithms)g Fp(whic)o(h)f(can:)75 1321 y(\(1\))21 b(pro)o(v)o(e)13 b(or)h(dispro)o(v)o(e) g(that)g(the)h(primitiv)o(e)c(of)j Fh(f)t Fp(\()p Fh(z)r Fp(\))h(can)f(b)q(e) g(expressed)j(with)c(elemen)o(tary)g(functions;)75 1371 y(\(2\))21 b(\014nd)14 b(a)f(solution)g(to)h(problem)e(\(1\).)0 1436 y(It)i(is)g(these)h (tec)o(hniques)g(that)f(w)o(e)g(w)o(an)o(t)g(to)f(in)o(tro)q(duce)i(here.)50 1488 y(Our)d(main)e(in)o(terest)j(will)d(b)q(e)i(the)g(in)o(tegration)f(of)g (rational)g(functions)g(and)h(of)f(purely)h(transcenden)o(tal)h(functions)f (\(this)0 1538 y(will)j(b)q(e)i(made)e(more)g(precise)j(later\).)25 b(W)m(e)16 b(will)f(fo)q(cus)h(on)g(the)h(in)o(tegration)f(of)f(rational)g (functions)i(\014rst,)g(as)f(the)h(main)0 1587 y(algorithm)11 b(can)i(b)q(e)h(deriv)o(ed)g(\(in)f(structure\))j(from)11 b(this)j(basic)f (case.)19 b(W)m(e)13 b(consider)h(that)g(w)o(e)f(ha)o(v)o(e)g(some)g(basic)g (to)q(ols)g(\\at)0 1637 y(hand",)h(whic)o(h)g(are)h(common)d(to)i(sym)o(b)q (olic)f(soft)o(w)o(ares.)20 b(This)14 b(to)q(olkit)g(includes)h(gcd)f(op)q (erations,)h(extended)h(Euclidean)0 1687 y(algorithm,)11 b(and)i(some)g (linear)h(algebra)f(tec)o(hniques.)747 1796 y Fq(2.)24 b(Rational)14 b(F)l(unctions)50 1872 y Fp(W)m(e)f(w)o(an)o(t)f(to)h(solv)o(e)h(the)f(follo) o(wing)e(problem:)16 b(giv)o(en)d Fh(F)k Fg(2)11 b Fb(Q)p Fp(\()p Fh(z)r Fp(\),)g(w)o(e)i(w)o(an)o(t)g(to)g(\014nd)1390 1838 y Fe(R)1425 1872 y Fh(F)6 b Fp(.)17 b(As)d(a)f(preliminary)e(step,)j(w)o(e)0 1922 y(can)g(write)868 1992 y Fh(F)j Fp(=)12 b Fh(P)j Fp(+)1044 1964 y Fh(Q)p 1044 1983 33 2 v 1044 2021 a(R)0 2076 y Fp(where)h Fh(P)q(;)7 b(Q;)g(R)14 b Fp(are)h(p)q(olynomials,)d Fh(Q)j Fp(and)g Fh(R)g Fp(are)g(coprime)f(and)h(deg)8 b Fh(Q)14 b(<)f Fp(deg)8 b Fh(R)p Fp(.)22 b(The)15 b(p)q(olynomial)d Fh(P)20 b Fp(can)c(b)q(e)f(easily)0 2126 y(in)o(tegrated,)f(so)g(the)g(remaining)e (task)i(is)g(to)g(compute)f(a)h(primitiv)o(e)d(of)j Fh(Q=R)p Fp(.)50 2217 y Fq(2.1.)24 b(Structural)13 b(approac)o(h.)19 b Fp(F)m(rom)12 b(the)j(theoretical)f(p)q(oin)o(t)f(of)h(view,)f(the)h(follo) o(wing)e(theorem)h(is)h(w)o(ell)f(kno)o(wn.)50 2308 y Fa(Theorem)j Fp(1)p Fa(.)21 b Fj(L)n(et)15 b Fh(R)c Fp(=)466 2277 y Fe(Q)505 2320 y Fi(i)p Fl(=1)p Fi(:::)o(n)611 2308 y Fp(\()p Fh(z)h Fg(\000)d Fh(a)721 2314 y Fi(i)735 2308 y Fp(\))751 2293 y Fi(n)772 2297 y Fc(i)802 2308 y Fj(b)n(e)14 b(the)h(factorisation)g(of)g Fh(R)p Fj(,)f(the)h(fr)n(action)f Fh(Q=R)h Fj(c)n(an)g(b)n(e)g(written)483 2388 y Fh(Q)p 483 2406 V 483 2444 a(R)532 2416 y Fp(=)598 2376 y Fe(X)576 2465 y Fi(i)p Fl(=1)p Fi(:::)o(n)757 2388 y Fh(B)788 2394 y Fi(i)p 692 2406 176 2 v 692 2444 a Fp(\()p Fh(z)c Fg(\000)f Fh(a)802 2450 y Fi(i)815 2444 y Fp(\))831 2432 y Fi(n)852 2436 y Fc(i)872 2416 y Fh(;)92 b(B)1007 2422 y Fi(i)1033 2416 y Fg(2)11 b Fb(Q)p Fp([)p Fh(z)r Fp(])o Fh(;)89 b Fp(deg)7 b Fh(B)1352 2422 y Fi(i)1378 2416 y Fh(<)12 b(n)1447 2422 y Fi(i)1461 2416 y Fh(:)0 2537 y Fj(We)j(c)n(an)h(r)n(ewrite)d(this)h(sum)601 2579 y Fh(Q)p 601 2598 33 2 v 601 2636 a(R)650 2607 y Fp(=)716 2568 y Fe(X)694 2656 y Fi(i)p Fl(=1)p Fi(:::)o(n)835 2568 y Fe(X)805 2656 y Fi(j)r Fl(=1)p Fi(:::)o(n)913 2660 y Fc(i)987 2579 y Fh(b)1005 2585 y Fi(i;j)p 937 2598 158 2 v 937 2636 a Fp(\()p Fh(z)e Fg(\000)d Fh(a)1047 2642 y Fi(i)1061 2636 y Fp(\))1077 2624 y Fi(j)1100 2607 y Fh(;)91 b(b)1221 2613 y Fi(i;j)1271 2607 y Fg(2)12 b Fb(Q)p Fh(:)954 2749 y Fp(59)p eop %%Page: 60 2 60 1 bop 0 33 a Fk(I)q(I)47 b(Sym)o(b)q(olic)16 b(Computation)50 141 y Fj(Then)f(the)g(primitive)f(of)g Fh(Q=R)h Fj(is)401 182 y Fe(Z)454 210 y Fh(Q)p 454 229 33 2 v 454 267 a(R)503 238 y Fp(=)569 199 y Fe(X)547 287 y Fi(i)p Fl(=1)p Fi(:::n)658 238 y Fh(b)676 244 y Fi(i;j)722 238 y Fp(log\()p Fh(z)c Fg(\000)f Fh(a)886 244 y Fi(i)899 238 y Fp(\))g Fg(\000)988 199 y Fe(X)966 287 y Fi(i)p Fl(=1)p Fi(:::n)1107 199 y Fe(X)1077 287 y Fi(j)r Fl(=2)p Fi(:::)o(n)1185 291 y Fc(i)1343 210 y Fh(b)1361 216 y Fi(i;j)p 1210 229 324 2 v 1210 267 a Fp(\()p Fh(j)h Fg(\000)f Fp(1\)\()p Fh(z)h Fg(\000)f Fh(a)1443 273 y Fi(i)1457 267 y Fp(\))1473 255 y Fi(j)r Ff(\000)p Fl(1)1538 238 y Fh(:)50 356 y Fp(This)16 b(giv)o(es)h(some)f(insigh)o(t)g(ab)q(out)g(the)i(result:)24 b(the)18 b(simple)d(factors)i(of)f(the)h(denominator)e(giv)o(e)i(logarithms)d (in)i(the)0 406 y(result,)j(while)f(m)o(ultiple)d(factors)k(giv)o(e)e (rational)g(functions.)30 b(Ho)o(w)o(ev)o(er,)19 b(this)f(form)o(ula)d(is)j (not)f(practicable,)i(for)f(man)o(y)0 456 y(reasons:)149 518 y(The)13 b(factorization)g(of)g Fh(R)g Fp(has)h(to)f(b)q(e)h(kno)o(wn,)f(and) g(it)g(is)g(not)h(easy)f(to)h(compute.)j(W)m(e)c(shall)g(see)h(that)g(it)149 568 y(is)f(not)h(needed)h(in)f(practice.)149 618 y(F)m(urthermore,)e(this)g (factorisation)g(giv)o(es)g(rise)h(to)f(algebraic)g(n)o(um)o(b)q(ers,)g (whose)h(use)h(is)e(exp)q(ensiv)o(e.)19 b(One)149 668 y(rule)14 b(in)f(computer)h(algebra)f(is)h(\\sta)o(y)f(in)h Fb(Q)d Fp(as)j(most)f(as)h (p)q(ossible".)50 730 y(Consider)g(the)h(follo)o(wing)c(rewriting)j(of)f Fh(Q)p Fp(:)820 829 y Fh(F)k Fp(=)11 b Fh(P)k Fp(+)998 801 y Fh(A)p 996 820 36 2 v 996 858 a(D)1045 829 y Fp(+)1092 801 y Fh(B)p 1092 820 34 2 v 1092 858 a(E)0 829 y Fp(\(2\))0 919 y(where)h Fh(A)p Fp(,)p Fh(B)r Fp(,)p Fh(D)q Fp(,)p Fh(E)r Fp(,)p Fh(P)k Fp(are)c(p)q(olynomial)o(s)d(in)i Fb(Q)p Fp([)o Fh(z)r Fp(],)d Fh(D)17 b Fp(is)e(square-free,)h(and)f Fh(E)i Fp(has)e(only)g(m)o(ultiple)d(factors.)23 b(W)m(e)14 b(w)o(an)o(t)h(to)0 968 y(deal)c(separately)h(with)e Fh(A=D)q Fp(,)i(whic)o(h)f(leads)g(to)g (logarithmic)d(terms,)j(and)g(with)g Fh(B)r(=E)i Fp(whic)o(h)e(leads)h(to)f (rational)e(functions.)50 1018 y(W)m(e)k(recall)h(that)g(the)h(square-free)g (decomp)q(osition)d(of)i(a)f(p)q(olynomial)e Fh(P)19 b Fp(is)14 b(the)g(decomp)q(osition)820 1094 y Fh(P)j Fp(=)12 b Fh(P)935 1100 y Fl(1)953 1094 y Fh(P)986 1077 y Fl(2)980 1104 y(2)1011 1094 y Fg(\001)7 b(\001)g(\001)e Fh(P)1099 1077 y Fi(m)1093 1104 y(m)0 1170 y Fp(where)19 b(the)f(p)q(olynomials)d Fh(P)461 1176 y Fl(1)479 1170 y Fh(;)7 b(:)g(:)g(:)e(;)i(P)599 1176 y Fi(m)647 1170 y Fp(are)18 b(pairwise)g(coprime,)f(and)g(eac)o(h)i Fh(P)1268 1176 y Fi(i)1299 1170 y Fp(is)e(square-free.)31 b(W)m(e)17 b(shall)g(call)g Fh(m)h Fp(the)0 1220 y Fj(highest)f(multiplicity)d Fp(of)i Fh(P)6 b Fp(.)23 b(This)16 b(decomp)q(osition)e(is)i(not)g(hard)f(to) h(compute,)g(using)f(di\013eren)o(tiation)h(of)f(p)q(olynomials)0 1269 y(and)f(gcd)g(op)q(erations)g(\(Consider)g(it)g(as)g(giv)o(en)f(in)h (the)g(basic)g(to)q(olkit\).)50 1319 y(In)g(fact)g(the)g(algorithm)d (computes)j(step)h(b)o(y)f(step:)769 1436 y Fh(F)j Fp(=)12 b Fh(Q)890 1419 y Ff(0)911 1436 y Fp(+)952 1365 y Fe( )1003 1397 y Fp(~)992 1408 y Fh(A)p 990 1426 36 2 v 1000 1459 a Fp(~)990 1470 y Fh(D)1031 1365 y Fe(!)1063 1374 y Ff(0)1084 1436 y Fp(+)1141 1397 y(~)1131 1408 y Fh(B)p 1131 1426 34 2 v 1141 1459 a Fp(~)1131 1470 y Fh(E)1169 1436 y(;)0 1556 y Fp(suc)o(h)i(that)g(the)g(highest)h(m)o (ultipli)o(cit)o(y)c(of)677 1546 y(~)667 1556 y Fh(E)16 b Fp(has)d(b)q(een)i (decreased,)h(so)e(that)f(w)o(e)h(ha)o(v)o(e)g(to)f(solv)o(e)h(the)g(same)f (problem)f(with)0 1606 y(lo)o(w)o(er)i(highest)g(m)o(ultiplicit)o(y)l(,)c(un) o(til)j(w)o(e)h(are)h(left)e(with)h(a)g(fraction)f(whose)h(denominator)f(is)g (square-free.)50 1656 y(This)k(pro)q(cess)i(of)d(decreasing)i(the)g(highest)f (m)o(ultiplicit)o(y)d(of)i(the)i(denominator)d(is)i(called)g(the)h Fj(Hermite)e(r)n(e)n(duction)p Fp(,)0 1706 y(whic)o(h)j(giv)o(es)g(the)g (rational)f(part,)i(and)f(the)h(logarithmic)c(part)j(\(for)g(a)g(square-free) h(denominator\))e(is)h(found)g(b)o(y)f(the)0 1755 y(metho)q(d)13 b(of)g(the)i Fj(R)n(othstein-T)m(r)n(ager)e Fp(resultan)o(t.)50 1832 y Fq(2.2.)24 b(The)14 b(Hermite)g(reduction.)k Fp(The)13 b(algorithm)d(dates)k(bac)o(k)f(to)g(Hermite.)k(W)m(e)12 b(sk)o(etc)o(h)i(a)f (mo)q(dern)f(v)o(ersion)h(due)0 1882 y(to)h(Mac)o(k.)k(W)m(e)13 b(presen)o(t)j(the)e(iterativ)o(e)g(step)h(of)e(the)i(metho)q(d.)0 1932 y Fq(Input)d Fh(P)s(=Q;)18 b Fp(deg)7 b Fh(P)17 b(<)12 b Fp(deg)c Fh(Q;)18 b Fp(gcd\()p Fh(P)q(;)7 b(Q)p Fp(\))k(=)h(1,)h(the)h (highest)h(m)o(ultipli)o(cit)o(y)c(of)i Fh(Q)h Fp(b)q(eing)g Fh(m)e(>)g Fp(1.)0 1982 y Fq(Output)h Fh(R;)7 b(B)r(;)g(E)15 b Fg(2)f Fb(Q)p Fp([)p Fh(X)s Fp(])e(suc)o(h)17 b(that)e Fh(P)s(=Q)f Fp(=)g(\()p Fh(R)p Fp(\))862 1966 y Ff(0)884 1982 y Fp(+)d Fh(B)r(=E)r Fp(,)k(the)h(highest)g(m)o(ultiplicit)o(y)c Fh(m)1519 1966 y Ff(0)1546 1982 y Fp(of)j Fh(E)j Fp(b)q(eing)d(lo)o(w)o(er)g(than)0 2031 y Fh(m)p Fp(.)104 2094 y Fq({)21 b Fp(Obtain)13 b(a)h(square-free)h (decomp)q(osition)e(of)g Fh(Q)p Fp(,)g(and)h(de\014ne)h Fh(G)e Fp(and)h Fh(G)1274 2079 y Fi(?)1306 2094 y Fp(as)g(sho)o(wn:)563 2170 y Fh(Q)e Fp(=)f Fh(Q)684 2176 y Fl(1)703 2170 y Fh(Q)736 2152 y Fl(2)736 2180 y(2)761 2170 y Fg(\001)c(\001)g(\001)f Fh(Q)850 2152 y Fi(m)850 2180 y(m)892 2170 y Fp(=)12 b Fh(Q)969 2176 y Fl(1)995 2170 y Fp(\()p Fh(Q)1044 2176 y Fl(2)1062 2170 y Fh(Q)1095 2176 y Fl(3)1121 2170 y Fg(\001)7 b(\001)g(\001)e Fh(Q)1209 2176 y Fi(m)1240 2170 y Fp(\))995 2197 y Fe(|)p 1014 2197 94 5 v 94 w({z)p 1146 2197 V 94 w(})1104 2237 y Fi(G)1130 2228 y Fc(?)1263 2170 y Fh(Q)1296 2176 y Fl(2)1315 2170 y Fh(Q)1348 2152 y Fl(2)1348 2180 y(3)1373 2170 y Fg(\001)i(\001)g(\001)e Fh(Q)1461 2152 y Fi(m)p Ff(\000)p Fl(1)1461 2180 y Fi(m)1263 2197 y Fe(|)p 1282 2197 99 5 v 99 w({z)p 1419 2197 V 99 w(})1386 2237 y Fi(G)104 2305 y Fq({)21 b Fp(Compute)12 b Fh(QG)394 2289 y Ff(0)405 2305 y Fh(=G)459 2289 y Fl(2)489 2305 y Fp(=)g Fh(Q)566 2311 y Fl(1)584 2305 y Fh(G)617 2289 y Fi(?)636 2305 y Fh(G)669 2289 y Ff(0)680 2305 y Fh(=G)p Fp(,)h(whic)o(h)h(can)g(b)q(e)g (seen)i(to)d(b)q(e)i(prime)e(to)g Fh(G)1409 2289 y Fi(?)1428 2305 y Fp(.)104 2354 y Fq({)21 b Fp(Using)13 b(the)i(extended)g(Euclidean)f (algorithm,)d(compute)i(the)i(Bezout)g(co)q(e\016cien)o(ts)g Fh(A;)7 b(B)16 b Fp(of)d Fh(P)6 b Fp(:)843 2452 y Fh(P)17 b Fp(=)12 b Fh(AQ)995 2458 y Fl(1)1018 2424 y Fh(G)1051 2409 y Fi(?)1070 2424 y Fh(G)1103 2409 y Ff(0)p 1018 2442 97 2 v 1050 2480 a Fh(G)1129 2452 y Fp(+)d Fh(B)r(G)1236 2435 y Fi(?)149 2539 y Fp(This)k(leads)h(to)803 2582 y Fh(P)p 803 2601 33 2 v 803 2639 a(Q)852 2610 y Fp(=)e Fg(\000)935 2552 y Fe(\022)971 2582 y Fh(A)p 970 2601 V 970 2639 a(G)1008 2552 y Fe(\023)1039 2560 y Ff(0)1059 2610 y Fp(+)1106 2582 y Fh(A)1137 2567 y Ff(0)1149 2582 y Fh(Q)1182 2588 y Fl(1)1209 2582 y Fp(+)e Fh(B)p 1106 2601 179 2 v 1153 2639 a(Q)1186 2645 y Fl(1)1205 2639 y Fh(G)1289 2610 y(:)952 2749 y Fm(60)p eop %%Page: 61 3 61 2 bop 1226 33 a Fk(In)o(tro)q(duction)16 b(to)f(sym)o(b)q(olic)h(in)o (tegration)104 141 y Fq({)21 b Fp(Return)13 b Fh(R)f Fp(=)g Fg(\000)p Fh(A=G)p Fp(,)g Fh(B)i Fp(=)e Fh(A)638 126 y Ff(0)650 141 y Fh(Q)683 147 y Fl(1)710 141 y Fp(+)c Fh(B)r Fp(,)14 b Fh(E)f Fp(=)f Fh(Q)930 147 y Fl(1)949 141 y Fh(G)p Fp(.)17 b(The)d(highest)g(m)o(ultipli)o(cit)o(y)d(of)h Fh(E)k Fp(is)d(no)o(w)g(at)g (most)g Fh(m)8 b Fg(\000)h Fp(1.)50 218 y Fa(Example.)22 b Fp(W)m(e)14 b(consider)g(the)h(follo)o(wing)c(function)657 319 y Fh(f)17 b Fp(=)742 291 y Fh(z)763 275 y Fl(7)791 291 y Fg(\000)10 b Fp(24)p Fh(z)896 275 y Fl(4)923 291 y Fg(\000)g Fp(4)p Fh(z)1007 275 y Fl(2)1035 291 y Fp(+)f(8)p Fh(z)i Fg(\000)f Fp(8)p 742 309 448 2 v 769 347 a Fh(z)790 335 y Fl(8)818 347 y Fp(+)f(6)p Fh(z)901 335 y Fl(6)929 347 y Fp(+)h(12)p Fh(z)1034 335 y Fl(4)1061 347 y Fp(+)g(8)p Fh(z)1145 335 y Fl(2)1206 319 y Fp(=)1255 291 y Fh(P)p 1255 309 33 2 v 1255 347 a(Q)0 415 y Fp(Using)k(this)g(algorithm:)728 491 y Fh(G)41 b Fp(=)h(gcd\()p Fh(Q;)7 b(Q)1039 474 y Ff(0)1050 491 y Fp(\))k(=)h Fh(z)1142 474 y Fl(5)1170 491 y Fp(+)e(4)p Fh(z)1254 474 y Fl(3)1282 491 y Fp(+)f(4)p Fh(z)657 554 y(Q)690 560 y Fl(1)709 554 y Fh(G)742 536 y Fi(?)802 554 y Fp(=)42 b Fh(Q=G)10 b Fp(=)i Fh(z)1038 536 y Fl(3)1066 554 y Fp(+)e(2)p Fh(z)563 616 y Fp(gcd\()p Fh(G;)d(G)726 599 y Fi(?)744 616 y Fp(\))42 b(=)g Fh(z)897 599 y Fl(2)925 616 y Fp(+)9 b(2)709 678 y Fh(Q)742 684 y Fl(1)802 678 y Fp(=)42 b(\()p Fh(Q)925 684 y Fl(1)943 678 y Fh(G)976 661 y Fi(?)995 678 y Fp(\))p Fh(=)p Fp(\()p Fh(G=)7 b Fp(gcd\()p Fh(G;)g(G)1272 661 y Ff(0)1282 678 y Fp(\)\))12 b(=)g(1)560 740 y Fh(Q)593 746 y Fl(1)611 740 y Fh(G)644 723 y Fi(?)663 740 y Fh(G)696 723 y Ff(0)707 740 y Fh(=G)41 b Fp(=)h(5)p Fh(z)918 723 y Fl(2)945 740 y Fp(+)10 b(2)p Fh(:)50 817 y Fp(By)k(Bezout,)h(w)o(e)f (ha)o(v)o(e)f Fh(P)k Fp(=)12 b Fg(\000)p Fp(\(8)p Fh(z)601 802 y Fl(2)629 817 y Fp(+)e(4\)\(5)p Fh(z)766 802 y Fl(2)794 817 y Fp(+)f(2\))g(+)h(\()p Fh(z)960 802 y Fl(4)988 817 y Fg(\000)f Fp(2)p Fh(z)1071 802 y Fl(2)1099 817 y Fp(+)h(16)p Fh(z)h Fp(+)e(4\)\()p Fh(z)1328 802 y Fl(3)1356 817 y Fp(+)h(2)p Fh(z)r Fp(\).)18 b(Th)o(us)614 926 y Fh(f)e Fp(=)694 868 y Fe(\022)785 898 y Fp(8)p Fh(z)827 883 y Fl(2)855 898 y Fp(+)10 b(4)p 730 917 244 2 v 730 955 a Fh(z)751 943 y Fl(5)778 955 y Fp(+)g(4)p Fh(z)862 943 y Fl(3)890 955 y Fp(+)f(4)p Fh(z)978 868 y Fe(\023)1009 874 y Ff(0)1030 926 y Fp(+)1087 898 y Fh(z)1108 883 y Fl(4)1136 898 y Fg(\000)g Fp(2)p Fh(z)1219 883 y Fl(2)1247 898 y Fp(+)g(4)p 1076 917 V 1076 955 a Fh(z)1097 943 y Fl(5)1125 955 y Fp(+)h(4)p Fh(z)1209 943 y Fl(3)1236 955 y Fp(+)g(4)p Fh(z)1325 926 y(:)50 1025 y Fp(Again,)i(for)i(the)g(second)h(term,)e(the)i(algorithm)c(giv)o(es:) 848 1102 y Fh(G)41 b Fp(=)h Fh(z)1017 1085 y Fl(2)1045 1102 y Fp(+)10 b(2)778 1164 y Fh(Q)811 1170 y Fl(1)829 1164 y Fh(G)862 1147 y Fi(?)922 1164 y Fp(=)42 b Fh(Q=G)11 b Fp(=)h Fh(z)1159 1147 y Fl(3)1187 1164 y Fp(+)d(2)p Fh(z)683 1226 y Fp(gcd)q(\()p Fh(G;)e(G)847 1209 y Fi(?)865 1226 y Fp(\))41 b(=)h(1)829 1289 y Fh(Q)862 1295 y Fl(1)922 1289 y Fp(=)g Fh(z)680 1351 y(Q)713 1357 y Fl(1)731 1351 y Fh(G)764 1334 y Fi(?)783 1351 y Fh(G)816 1334 y Ff(0)827 1351 y Fh(=G)f Fp(=)h(2)p Fh(z)1038 1334 y Fl(2)1057 1351 y Fh(:)50 1428 y Fp(W)m(e)13 b(get)h(the)h(Bezout)g(co)q (e\016cien)o(ts)g Fh(z)633 1413 y Fl(4)661 1428 y Fg(\000)10 b Fp(2)p Fh(z)745 1413 y Fl(2)772 1428 y Fp(+)g(4)h(=)h Fg(\000)p Fp(3\(2)p Fh(z)1001 1413 y Fl(2)1020 1428 y Fp(\))d(+)h(\()p Fh(z)1124 1413 y Fl(2)1152 1428 y Fp(+)f(2\)\()p Fh(z)1267 1413 y Fl(2)1295 1428 y Fp(+)h(2\).)18 b(Ev)o(en)o(tually:)645 1537 y Fh(f)e Fp(=)725 1478 y Fe(\022)816 1509 y Fp(8)p Fh(z)858 1494 y Fl(2)886 1509 y Fp(+)10 b(4)p 761 1527 V 761 1565 a Fh(z)782 1553 y Fl(5)810 1565 y Fp(+)f(4)p Fh(z)893 1553 y Fl(3)921 1565 y Fp(+)g(4)p Fh(z)1018 1537 y Fp(+)1110 1509 y(3)p 1065 1527 112 2 v 1065 1565 a Fh(z)1086 1553 y Fl(2)1114 1565 y Fp(+)g(2)1181 1478 y Fe(\023)1212 1485 y Ff(0)1233 1537 y Fp(+)1279 1509 y(1)p 1279 1527 22 2 v 1279 1565 a Fh(z)50 1636 y Fq(2.3.)24 b(The)14 b(Hurwitz-Ostrograds)o(ky)e(metho)q(d.)19 b Fp(It)13 b(is)f(p)q(ossible)h(to)f(compute)f(directly)i(the)g(result)g(b)o (y)f(the)h(follo)o(w-)0 1686 y(ing)g(remark.)k(The)e(aim)c(is)j(to)g(compute) 738 1738 y Fh(P)p 738 1756 33 2 v 738 1794 a(Q)787 1766 y Fp(=)831 1707 y Fe(\022)861 1766 y Fg(\000)899 1738 y Fh(A)p 898 1756 V 898 1794 a(G)936 1707 y Fe(\023)967 1716 y Ff(0)988 1766 y Fp(+)1107 1738 y Fh(C)p 1034 1756 179 2 v 1034 1794 a(Q)1067 1800 y Fl(1)1093 1794 y Fg(\001)7 b(\001)g(\001)e Fh(Q)1181 1800 y Fi(m)0 1851 y Fp(i.e.)681 1917 y Fh(P)16 b Fp(=)c Fh(AQ)832 1923 y Fl(1)856 1888 y Fh(G)889 1873 y Fi(?)908 1888 y Fh(G)941 1873 y Ff(0)p 856 1907 97 2 v 888 1945 a Fh(G)966 1917 y Fg(\000)e Fh(A)1039 1899 y Ff(0)1050 1917 y Fh(Q)1083 1923 y Fl(1)1102 1917 y Fh(G)1135 1899 y Fi(?)1163 1917 y Fp(+)f Fh(C)s(G)0 1995 y Fp(with)k(the)g(conditions:)k(deg)8 b Fh(A)k(<)f Fp(deg)d Fh(G)p Fp(,)k(deg)c Fh(C)14 b(<)e Fp(deg)q(\()p Fh(Q)931 2001 y Fl(1)950 1995 y Fh(G)983 1979 y Fi(?)1001 1995 y Fp(\).)18 b(This)13 b(is)f(a)h(linear)f(system)h(in)f(the)i(co)q(e\016cien)o(ts)g(of)e Fh(A)h Fp(and)0 2044 y Fh(C)s Fp(,)g(whic)o(h)h(can)g(b)q(e)g(directly)h (solv)o(ed)e(b)o(y)h(a)g(linear)f(solv)o(er)h(\(again)f(in)g(the)i(basic)f (mac)o(hinery)e(of)i(the)g(to)q(olkit\).)50 2122 y Fq(2.4.)24 b(The)17 b(logarithmi)o(c)d(part.)20 b Fp(No)o(w)15 b(it)f(remains)g(to)h (solv)o(e)f(the)i(problem)d(for)i(a)f(fraction)h Fh(f)j Fp(=)13 b Fh(P)s(=Q)p Fp(,)h(where)i Fh(Q)f Fp(is)0 2172 y(square-free.)k(A)o(t)14 b(this)g(p)q(oin)o(t)g(t)o(w)o(o)f(di\016culties)h(m)o(ust)f(b)q(e)h(p)q(oin) o(ted)g(out.)k(Consider)c(the)h(example)572 2277 y Fh(f)i Fp(=)713 2249 y(5)p Fh(z)755 2233 y Fl(4)783 2249 y Fp(+)9 b(60)p Fh(z)887 2233 y Fl(3)915 2249 y Fp(+)g(255)p Fh(z)1040 2233 y Fl(2)1068 2249 y Fp(+)g(450)p Fh(z)i Fp(+)e(275)p 657 2267 704 2 v 657 2305 a Fh(z)678 2293 y Fl(5)706 2305 y Fp(+)h(15)p Fh(z)811 2293 y Fl(4)838 2305 y Fp(+)g(85)p Fh(z)943 2293 y Fl(3)970 2305 y Fp(+)g(225)p Fh(z)1096 2293 y Fl(2)1123 2305 y Fp(+)g(274)p Fh(z)g Fp(+)g(120)1366 2277 y Fh(;)-1378 b Fp(\(3\))0 2368 y(whose)14 b(primitiv)o(e)e(is)312 2434 y(25)p 312 2452 42 2 v 312 2490 a(24)365 2462 y(log\()p Fh(z)f Fp(+)f(1\))f(+)599 2434 y(5)p 599 2452 21 2 v 599 2490 a(6)632 2462 y(log\()p Fh(z)i Fp(+)f(2\))f(+)866 2434 y(5)p 866 2452 V 866 2490 a(4)899 2462 y(log)o(\()p Fh(z)j Fp(+)d(3\))h(+)1133 2434 y(5)p 1133 2452 V 1133 2490 a(6)1166 2462 y(log)o(\()p Fh(z)i Fp(+)d(4\))g(+)1400 2434 y(25)p 1400 2452 42 2 v 1400 2490 a(24)1453 2462 y(log\()p Fh(z)i Fp(+)f(5\))p Fh(;)0 2550 y Fp(giv)o(en)15 b(in)h(expanded)g(form.)22 b(If)15 b(the)i(factors)f(where)h(factored)f(in)f(simple)g(logarithm,)e(it)i (w)o(ould)g(b)q(e)i(the)f(logarithm)d(of)i(a)0 2600 y(degree)e(120)d(p)q (olynomial,)e(with)j(co)q(e\016cien)o(ts)h(of)f(order)h(10)916 2585 y Fl(68)951 2600 y Fp(.)17 b(So)11 b(the)h(problem)e(of)g(the)i(gro)o (wth)f(of)g(in)o(termediate)f(n)o(um)o(b)q(ers)0 2650 y(is)k(presen)o(t)h (here.)952 2749 y Fm(61)p eop %%Page: 62 4 62 3 bop 0 33 a Fk(I)q(I)47 b(Sym)o(b)q(olic)16 b(Computation)50 141 y Fp(The)e(other)g(di\016cult)o(y)f(is)g(that)h(algebraic)f(n)o(um)o(b)q (ers)g(ma)o(y)f(b)q(e)i(required)g(in)f(the)i(solution,)d(as)i(sho)o(wn)f(b)o (y)h(the)g(example:)722 191 y Fe(Z)809 219 y Fh(dz)p 775 238 112 2 v 775 276 a(z)796 264 y Fl(2)824 276 y Fg(\000)c Fp(2)903 248 y(=)952 185 y Fg(p)p 986 185 21 2 v 986 219 a Fp(2)p 952 238 56 2 v 969 276 a(4)1019 248 y(log)1085 219 y Fh(z)h Fg(\000)1156 185 y(p)p 1191 185 21 2 v 34 x Fp(2)p 1085 238 128 2 v 1085 281 a Fh(z)g Fp(+)1156 246 y Fg(p)p 1191 246 21 2 v 35 x Fp(2)1217 248 y Fh(:)50 348 y Fp(Then)j(the)h(problem)d(is)i(to)g(compute)f(in)g(the)i (extension)g(of)e Fb(Q)e Fp(of)i(lo)o(w)o(est)h(degree.)50 399 y(The)g(metho)q(d)f(of)g(Rothstein-T)m(rager)h(giv)o(es)g(a)g(go)q(o)q(d) f(solution.)k(Consider)764 496 y Fh(f)f Fp(=)849 468 y Fh(P)p 849 486 33 2 v 849 524 a(Q)898 496 y Fp(=)964 456 y Fe(X)942 545 y Fi(i)p Fl(=1)p Fi(:::)o(n)1096 468 y Fh(a)1118 474 y Fi(i)p 1058 486 113 2 v 1058 524 a Fh(z)11 b Fg(\000)f Fh(\013)1157 530 y Fi(i)1175 496 y Fh(:)104 613 y Fq({)21 b Fp(One)15 b(can)f(c)o(hec)o(k) h(that)f Fh(a)537 619 y Fi(i)563 613 y Fp(=)f Fh(P)6 b Fp(\()p Fh(\013)684 619 y Fi(i)697 613 y Fp(\))p Fh(=Q)767 597 y Ff(0)778 613 y Fp(\()p Fh(\013)821 619 y Fi(i)835 613 y Fp(\),)14 b(th)o(us)g(the)h(p) q(olynomials)c Fh(Q)j Fp(and)g Fh(P)h Fg(\000)10 b Fh(a)1505 619 y Fi(i)1519 613 y Fh(Q)1552 597 y Ff(0)1578 613 y Fp(ha)o(v)o(e)k(a)g (common)d(ro)q(ot)149 662 y Fh(\013)176 668 y Fi(i)189 662 y Fp(.)23 b(So)16 b Fh(a)306 668 y Fi(i)335 662 y Fp(is)g(a)f(ro)q(ot)h(of)f Fh(R)g Fp(=)f(Res)714 668 y Fi(z)733 662 y Fp(\()p Fh(Q;)7 b(P)15 b Fg(\000)c Fh(tQ)934 647 y Ff(0)945 662 y Fp(\))16 b(whic)o(h)g(b)q(elongs)g(to)f Fb(Q)p Fp([)p Fh(t)p Fp(].)20 b(\(Res)1489 668 y Fi(z)1523 662 y Fp(stands)d(for)e(the)i Fj(r)n(esultant)149 712 y(in)d Fh(z)r Fp(,)g(another)g(basic)f(to)q(ol,)g (whic)o(h)h(giv)o(es)f(algebraic)g(conditions)h(on)f(co)q(e\016cien)o(ts)i (for)e(t)o(w)o(o)g(p)q(olynomials)e(in)i Fh(z)j Fp(to)149 762 y(ha)o(v)o(e)d(a)h(common)d(ro)q(ot.\))104 812 y Fq({)21 b Fp(Then)14 b Fh(\013)284 818 y Fi(i)310 812 y Fp(is)g(a)f(ro)q(ot)h(of)f Fh(G)555 818 y Fi(i)580 812 y Fp(=)e(gcd)q(\()p Fh(Q;)c(P)13 b Fg(\000)c Fh(a)857 818 y Fi(i)871 812 y Fh(Q)904 797 y Ff(0)915 812 y Fp(\))14 b(whic)o(h)f(b)q(elongs)h(to)f Fb(Q)p Fp(\()p Fh(a)1335 818 y Fi(i)1346 812 y Fp(\)[)p Fh(z)r Fp(].)18 b(These)c(ro)q(ots)g (can)g(b)q(e)g(collected)149 862 y(for)f(eac)o(h)h Fh(a)327 868 y Fi(i)341 862 y Fp(:)k(for)c(eac)o(h)g(suc)o(h)h Fh(a)644 868 y Fi(i)658 862 y Fp(,)e(the)h(answ)o(er)h(is)707 946 y Fh(a)729 952 y Fi(i)815 906 y Fe(X)750 997 y Fi(\013)772 1001 y Fc(j)787 997 y Ff(j)p Fi(G)823 1001 y Fc(i)836 997 y Fl(\()p Fi(\013)871 1001 y Fc(j)886 997 y Fl(\)=0)948 946 y Fp(log)o(\()p Fh(z)d Fg(\000)d Fh(\013)1116 952 y Fi(j)1133 946 y Fp(\))j(=)g Fh(a)1227 952 y Fi(i)1248 946 y Fp(log)o(\()p Fh(G)1350 952 y Fi(i)1364 946 y Fp(\))p Fh(:)104 1071 y Fq({)21 b Fp(The)14 b(\014nal)f(answ)o(er)i(is)692 1079 y Fe(Z)746 1107 y Fh(P)p 746 1126 33 2 v 746 1164 a(Q)795 1135 y Fp(=)878 1096 y Fe(X)839 1187 y Fi(a)p Ff(j)p Fi(R)p Fl(\()p Fi(a)p Fl(\)=0)985 1135 y Fh(a)7 b Fp(log)o(\(gcd)q(\()p Fh(Q;)g(P)14 b Fg(\000)9 b Fh(aQ)1351 1118 y Ff(0)1363 1135 y Fp(\)\))p Fh(:)0 1247 y Fp(It)14 b(m)o(ust)f(b)q(e)h(p)q(oin)o(ted)g(out)g(that)g(this)g(is)g(the)g (expression)h(of)e(the)i(primitiv)o(e)d(in)h(the)i(lo)o(w)o(est)e(degree)j (extension)e(of)f Fb(Q)p Fp(.)50 1326 y Fa(Example.)22 b Fp(W)m(e)10 b(consider)g(the)h(function)f Fh(f)k Fp(giv)o(en)c(b)o(y)f(\(3\).W)m(e)g(ha)o (v)o(e)h(Res)1204 1332 y Fi(z)1223 1326 y Fp(\()p Fh(Q;)d(P)g Fg(\000)q Fh(tQ)1406 1311 y Ff(0)1418 1326 y Fp(\))k(=)h(\(5)q Fg(\000)q Fp(4)p Fh(t)p Fp(\)\(6)p Fh(t)q Fg(\000)q Fp(5\))1735 1311 y Fl(2)1755 1326 y Fp(\(24)p Fh(t)q Fg(\000)q Fp(25\))1920 1311 y Fl(2)1938 1326 y Fh(;)0 1376 y Fp(and)456 1409 y Fe(Z)505 1465 y Fh(f)k Fp(=)590 1437 y(5)p 590 1456 21 2 v 590 1494 a(4)623 1465 y(log)o(\()p Fh(z)c Fp(+)d(3\))g(+)897 1426 y Fe(X)852 1520 y Fi(\013)p Ff(2f)918 1509 y Fd(25)p 918 1514 29 2 v 918 1530 a(24)951 1520 y Fi(;)966 1509 y Fd(5)p 966 1514 15 2 v 966 1530 a(6)985 1520 y Ff(g)1009 1465 y Fh(\013)e Fp(log)o(\()p Fh(z)1133 1448 y Fl(2)1161 1465 y Fp(+)j(6)p Fh(z)h Fp(+)e(20)g Fg(\000)h Fp(72)1435 1437 y Fh(\013)p 1435 1456 27 2 v 1438 1494 a Fp(5)1466 1465 y(\))p Fh(:)722 1612 y Fq(3.)24 b(Elemen)o(tary)14 b(functions)50 1687 y(3.1.)24 b(De\014nitio)o(n.)18 b Fp(The)c(classical)g(de\014nition)f(of)h(elemen)o (tary)f(functions,)g(after)i(Liouville's)d(w)o(ork,)h(is:)50 1766 y Fa(Definition)j Fp(1)p Fa(.)21 b Fp(Let)16 b Fh(K)i Fp(b)q(e)e(a)f(di\013eren)o(tial)h(\014eld,)f Fh(y)i Fp(is)f(said)f(to)g(b)q (e)h Fj(elementary)g Fp(o)o(v)o(er)f Fh(K)k Fp(if)14 b Fh(K)s Fp(\()p Fh(y)q Fp(\))j(and)e Fh(K)k Fp(ha)o(v)o(e)c(the)0 1816 y(same)e(\014eld)h(of)f(constan)o(ts,)i(and)e(if)75 1879 y(\(1\))21 b(either)14 b Fh(y)i Fp(is)e(algebraic)f(o)o(v)o(er)h Fh(K)s Fp(,)75 1929 y(\(2\))21 b(or)13 b(there)j(exists)e Fh(f)j Fg(2)11 b Fh(K)17 b Fp(suc)o(h)d(that)g Fh(f)755 1914 y Ff(0)779 1929 y Fp(=)e Fh(f)t(y)868 1914 y Ff(0)895 1929 y Fp(\(i.e.)h Fh(y)j Fp(is)e(some)f(kind)g(of)g(logarithm\),)75 1979 y(\(3\))21 b(or)13 b(there)j(exists)e Fh(f)j Fg(2)11 b Fh(K)17 b Fp(suc)o(h)d(that)g Fh(y)752 1964 y Ff(0)776 1979 y Fp(=)e Fh(y)q(f)865 1964 y Ff(0)892 1979 y Fp(\(i.e.)h Fh(y)j Fp(is)e(an)f(exp)q(onen)o(tial\).)50 2058 y(No)o(w)i(the)h(follo)o(wing)c(theorem)j(giv)o(es)g(an)g(imp)q(ortan)o (t)e(c)o(haracterization)j(of)f(functions)g(whic)o(h)g(admit)f(an)h(elemen)o (tary)0 2107 y(primitiv)o(e:)50 2188 y Fa(Theorem)h Fp(2)g(\()p Fa(Liouville)p Fp(\))p Fa(.)23 b Fj(L)n(et)18 b Fh(K)k Fj(b)n(e)e(a)f (di\013er)n(ential)f(\014eld,)i(and)g Fh(C)i Fj(b)n(e)d(the)g(\014eld)g(of)g (c)n(onstants)g(of)g Fh(K)s Fj(,)h Fh(f)k Fg(2)19 b Fh(K)0 2238 y Fj(admits)c(an)g(elementary)g(primitive)f(over)g Fh(K)k Fj(if)d(and)g(only)g(if)g Fh(f)k Fj(c)n(an)d(b)n(e)f(written)820 2347 y Fh(f)h Fp(=)c Fh(v)921 2330 y Ff(0)942 2347 y Fp(+)999 2295 y Fi(m)984 2308 y Fe(X)987 2396 y Fi(i)p Fl(=1)1051 2347 y Fh(c)1069 2353 y Fi(i)1088 2319 y Fh(u)1112 2304 y Ff(0)1112 2330 y Fi(i)p 1088 2338 38 2 v 1088 2376 a Fh(u)1112 2382 y Fi(i)0 2469 y Fj(wher)n(e)i Fh(v)f Fg(2)f Fh(K)s Fj(,)i Fh(c)273 2475 y Fi(i)299 2469 y Fg(2)p 338 2436 33 2 v 11 w Fh(C)s Fj(,)g Fh(u)422 2475 y Fi(i)447 2469 y Fg(2)d Fh(K)s Fj(.)50 2550 y Fp(The)j(in)o(tegration)f(algorithm)e(in)i(the)h(case)h(of)e(algebraic)g (functions)h(relies)g(on)f(a)h(m)o(uc)o(h)e(more)h(di\016cult)g(theory)m(,)g (whic)o(h)0 2600 y(b)q(orro)o(ws)i(some)e(to)q(ols)i(from)d(algebraic)i (geometry)m(.)19 b(It)14 b(is)h(not)f(within)g(the)h(scop)q(e)g(of)f(this)h (talk)e(to)i(explain)e(these.)22 b(So)14 b(w)o(e)0 2650 y(shall)f(restrict)j (ourselv)o(es)e(to)g(transcenden)o(tal)h(functions,)f(whic)o(h)g(are)g (amazingly)d(simpler)i(to)g(deal)h(with.)952 2749 y Fm(62)p eop %%Page: 63 5 63 4 bop 1226 33 a Fk(In)o(tro)q(duction)16 b(to)f(sym)o(b)q(olic)h(in)o (tegration)50 141 y Fq(3.2.)24 b(The)15 b(strategy)f(of)h(Risc)o(h's)f (algorithm;)f(transcenden)o(tal)f(case.)21 b Fp(The)14 b(Risc)o(h)g (algorithm)c(app)q(ears)15 b(to)e(b)q(e)0 191 y(a)g(sligh)o(t)f (generalisation)h(of)f(the)i(metho)q(d)e(w)o(e)i(ha)o(v)o(e)e(describ)q(ed)j (here)g(for)d(the)i(case)g(of)f(rational)f(functions.)17 b(W)m(e)c(giv)o(e)g (here)0 241 y(Bronstein's)i(v)o(ersion)f(of)f(Risc)o(h's)h(algorithm,)c(in)k (the)g(transcenden)o(tal)i(case:)104 304 y Fq({)21 b Fp(Find)d(a)h(to)o(w)o (er)g(of)g(extensions)h(of)e Fb(Q)f Fp(giv)o(en)h(b)o(y)h Fh(\022)967 310 y Fl(1)1006 304 y Fp(=)i Fh(z)r(;)7 b(\022)1118 310 y Fl(2)1136 304 y Fh(;)g(:)g(:)g(:)e(;)i(\022)1248 310 y Fi(k)1288 304 y Fp(suc)o(h)19 b(that)h Fh(\022)1501 310 y Fi(i)1534 304 y Fp(is)f(elemen)o(tary)f(o)o(v)o(er)h(the)149 354 y(previous)14 b(\014eld)g Fb(Q)p Fp(\()p Fh(\022)474 360 y Fl(1)490 354 y Fh(;)7 b(:)g(:)g(:)e(;)i(\022)602 360 y Fi(i)p Ff(\000)p Fl(1)658 354 y Fp(\))14 b(for)g(ev)o(ery)g Fh(i)p Fp(,)g(and)g(suc)o(h)g(that)g Fh(f)j Fg(2)11 b Fb(Q)p Fp(\()p Fh(\022)1311 360 y Fl(1)1327 354 y Fh(;)c(:)g(:)g(:)e(;)i(\022)1439 360 y Fi(k)1460 354 y Fp(\).)18 b(W)m(e)13 b(note)h Fh(\022)g Fp(=)d Fh(\022)1764 360 y Fi(k)1785 354 y Fp(.)104 404 y Fq({)21 b Fp(W)m(rite)13 b Fh(f)j Fp(=)c Fh(A)p Fp(\()p Fh(\022)q Fp(\))p Fh(=D)q Fp(\()p Fh(\022)q Fp(\),)k(with)d Fh(A;)7 b(D)13 b Fg(2)e Fb(Q)p Fp(\()p Fh(\022)863 410 y Fl(1)880 404 y Fh(;)c(:)g(:)g(:)t(;)g(\022)991 410 y Fi(k)q Ff(\000)p Fl(1)1054 404 y Fp(\).)104 454 y Fq({)21 b Fp(Split)d Fh(D)j Fp(in)o(to)d Fh(D)23 b Fp(=)d Fh(D)540 460 y Fi(s)558 454 y Fh(D)592 460 y Fi(n)615 454 y Fp(,)h(where)f Fh(D)807 460 y Fi(s)844 454 y Fp(is)g(the)f(pro)q(duct)i(of)d(the)i (square-free)h(factors)f Fh(F)k Fp(of)19 b Fh(D)i Fp(suc)o(h)f(that)149 503 y(gcd\()p Fh(F)q(;)7 b(F)307 488 y Ff(0)317 503 y Fp(\))12 b Fg(6)p Fp(=)g(1.)18 b(W)m(e)13 b(rewrite)i(the)f(function)g Fh(f)19 b Fp(as)14 b(follo)o(ws)733 603 y Fh(f)j Fp(=)11 b Fh(P)k Fp(+)911 575 y Fh(B)p 902 593 53 2 v 902 631 a(D)936 637 y Fi(s)968 603 y Fp(+)1027 575 y Fh(C)p 1014 593 57 2 v 1014 631 a(D)1048 637 y Fi(n)1088 603 y Fp(=)d Fh(f)1152 609 y Fi(p)1180 603 y Fp(+)e Fh(f)1242 609 y Fi(s)1269 603 y Fp(+)g Fh(f)1331 609 y Fi(n)1354 603 y Fh(:)149 699 y Fp(Here)15 b Fh(f)267 705 y Fi(p)300 699 y Fp(is)f(called)f(the)i(p)q(olynomial)10 b(part,)k Fh(f)867 705 y Fi(s)899 699 y Fp(is)g(the)g(sp)q(ecial)g(part,)g (and)f Fh(f)1349 705 y Fi(n)1386 699 y Fp(the)i(normal)d(part.)104 749 y Fq({)21 b Fp(A)15 b(more)e(general)j(case)f(of)g(Hermite's)f(reduction) i(is)f(applied,)f(to)g(eliminate)f(all)h(factors)h(of)g(the)g(denominators) 149 799 y(whose)f(highest)g(m)o(ultiplicit)o(y)d(is)i(greater)i(than)f(1.)k (This)c(giv)o(es)910 877 y Fh(f)j Fp(=)11 b Fh(f)1010 883 y Fi(p)1039 877 y Fp(+)f Fh(g)1102 860 y Ff(0)1123 877 y Fp(+)f Fh(h)149 955 y Fp(where)15 b(the)f(denominator)e(of)i Fh(h)f Fp(has)h(no)g(m)o(ultiple)e(factors.)104 1005 y Fq({)21 b Fp(A)d(Rothstein-T) m(rager)h(resultan)o(t)g(is)g(computed)f(to)g(\014nd)h(the)g(logarithmic)d (part)j(\(or)g(to)g(decide)g(it)g(do)q(es)g(not)149 1055 y(exists\).)104 1105 y Fq({)i Fp(The)d(p)q(olynomial)c(part)k Fh(f)570 1111 y Fi(p)607 1105 y Fp(is)g(in)o(tegrated)g(b)o(y)f(an)h(appropriate)f(metho)q (d,)h(whether)h Fh(y)g Fp(is)f(a)f(logarithm)e(or)j(an)149 1155 y(exp)q(onen)o(tial.)50 1218 y(It)c(is)h(understo)q(o)q(d)g(that)g(it)f (is)g(a)g(recursiv)o(e)i(metho)q(d,)e(since)h(an)f(e\013ectiv)o(e)i(metho)q (d)e(for)g(the)h(case)g Fh(k)f Fp(=)f(1)h(\(i.e.)f(a)h(simple)0 1268 y(extension\))e(giv)o(es)e(a)h(metho)q(d)f(for)g(the)i(general)f(case,)h (where)g(computations)d(are)i(p)q(erformed)g(is)g(some)e(to)o(w)o(er)j(of)e (extension,)0 1318 y(where)15 b(arithmetic)e(is)h(e\013ectiv)o(e,)g(and)g (primitiv)o(es)e(recursiv)o(ely)j(computed.)50 1398 y Fq(3.3.)24 b(More)19 b(details.)f Fp(No)o(w)f(w)o(e)g(giv)o(e)f(details)g(ab)q(out)h (the)g(main)e(steps)j(of)e(Risc)o(h's)h(algorithm.)24 b(F)m(or)16 b(constructing)0 1448 y(the)e(extension)h(to)o(w)o(er,)f(and)f(\014nding)h (the)g(primitiv)o(e)e(of)h(the)i(sp)q(ecial)f(part,)f(see)j(the)e (references.)50 1529 y(3.3.1.)39 b(The)18 b(Hermite)f(reduction)h(W)m(e)f (are)g(concerned)j(with)c(the)i(normal)e(part)h Fh(A=D)1470 1535 y Fi(n)1493 1529 y Fp(,)g(whic)o(h)h(w)o(e)f(w)o(ould)g(lik)o(e)f(to)0 1579 y(rewrite)g Fh(A=D)228 1585 y Fi(n)265 1579 y Fp(=)d Fh(g)331 1564 y Ff(0)353 1579 y Fp(+)e Fh(P)k Fp(+)c Fh(B)r(=Q)568 1585 y Fi(n)590 1579 y Fp(,)k(where)i Fh(Q)772 1585 y Fi(n)809 1579 y Fp(has)e(no)g(m)o(ultiple)e(factors.)22 b(As)16 b(b)q(efore)g(w)o(e)f (describ)q(e)i(a)e(single)f(step)j(of)0 1629 y(the)d(metho)q(d,)f(for)h (reducing)g(the)h(highest)f(m)o(ultiplicit)o(y)c(at)k(least)g(b)o(y)g(one.)50 1679 y(Let)i Fh(V)159 1664 y Fi(k)q Fl(+1)237 1679 y Fp(b)q(e)g(the)g (highest)g(exp)q(onen)o(t)g(in)f(the)h(square-free)h(decomp)q(osition)d(of)h Fh(D)1386 1685 y Fi(n)1408 1679 y Fp(,)h(that)f(is)g Fh(D)1604 1685 y Fi(n)1641 1679 y Fp(=)g Fh(U)5 b(V)1754 1664 y Fi(k)q Fl(+1)1817 1679 y Fp(,)15 b(where)0 1729 y Fh(V)23 b Fp(is)14 b(square-free,)h(gcd\()p Fh(U;)7 b(V)i Fp(\))j(=)g(1)i(and)f(the)i(highest)f (m)o(ultiplicit)o(y)c(of)k Fh(U)k Fp(is)c(less)h(than)f Fh(k)q Fp(.)50 1779 y(W)m(e)f(seek)i(t)o(w)o(o)f(p)q(olynomial)o(s)e Fh(G;)7 b(H)13 b Fg(2)e Fh(K)s Fp([)p Fh(\022)q Fp(],)j(suc)o(h)g(that)g(deg) 8 b Fh(G)j(<)h Fp(deg)c Fh(V)23 b Fp(and)784 1860 y Fh(A)p 735 1879 129 2 v 735 1917 a(U)5 b(V)801 1905 y Fi(k)q Fl(+1)880 1888 y Fp(=)924 1830 y Fe(\022)970 1860 y Fh(G)p 960 1879 54 2 v 960 1917 a(V)993 1905 y Fi(k)1019 1830 y Fe(\023)1049 1838 y Ff(0)1070 1888 y Fp(+)1141 1860 y Fh(H)p 1117 1879 87 2 v 1117 1917 a(U)g(V)1183 1905 y Fi(k)1208 1888 y Fh(:)0 1992 y Fp(\(Remem)o(b)q(er)16 b(that)h(our)h(goal)e(is)h(to)h(lo)o(w)o(er)f(the)h (highest)g(m)o(ultiplici)o(t)o(y)d(of)i(the)h(denominator.\))27 b(A)17 b(small)f(computation)0 2041 y(giv)o(es)e(the)g(relation)g Fh(A)d Fp(=)h Fh(U)5 b(V)k(G)511 2026 y Ff(0)532 2041 y Fg(\000)g Fh(k)q(U)c(V)663 2026 y Ff(0)674 2041 y Fh(G)k Fp(+)h Fh(V)f(H)s Fp(,)k(and)h(mo)q(dulo)e Fh(V)d Fp(,)14 b(w)o(e)g(ha)o(v)o(e)768 2139 y Fh(G)e Fp(=)f Fg(\000)928 2110 y Fh(A)p 893 2129 101 2 v 893 2167 a(k)q(U)5 b(V)983 2155 y Ff(0)1053 2139 y Fp(mo)q(d)10 b Fh(V)r(:)0 2231 y Fp(Again)g(w)o(e)i(pic)o(k)f(in)g(our)g(basic)g(to)q (olkit)f(the)i(extended)h(Euclidean)e(algorithm)e(to)i(compute)f Fh(A=k)q(U)5 b(V)1585 2216 y Ff(0)1608 2231 y Fp(mo)q(d)10 b Fh(V)f Fp(.)17 b Fh(H)d Fp(is)d(easily)0 2281 y(computed)i(from)g(the)h (data)g(of)f Fh(A;)7 b(U;)g(V)r(;)g(G;)g(k)q Fp(.)50 2360 y Fa(Example.)22 b Fp(Consider)602 2427 y Fh(f)16 b Fp(=)687 2398 y Fh(z)11 b Fg(\000)f Fp(tan\()p Fh(z)r Fp(\))p 687 2417 186 2 v 714 2456 a(tan)774 2441 y Fl(2)792 2456 y Fp(\()p Fh(z)r Fp(\))930 2427 y Fg(2)i Fb(Q)p Fp(\()p Fh(z)r Fp(\)\()p Fh(\022)q Fp(\))p Fh(;)k(\022)1158 2409 y Ff(0)1182 2427 y Fp(=)c(1)d(+)g Fh(\022)1317 2409 y Fl(2)1337 2427 y Fh(:)0 2518 y Fp(The)14 b(follo)o(wing)e(equation)h(is)h(to)g(b)q(e)g(solv)o(ed)g(for)f Fh(G;)7 b(H)14 b Fg(2)d Fb(Q)p Fp(\()p Fh(z)r Fp(\)[)p Fh(\022)q Fp(])p Fh(;)c Fp(deg)e Fh(G)11 b(<)h Fp(1:)788 2599 y Fh(z)g Fg(\000)d Fh(\022)p 788 2617 93 2 v 815 2655 a(\022)835 2643 y Fl(2)897 2627 y Fp(=)941 2568 y Fe(\022)977 2599 y Fh(G)p 977 2617 33 2 v 983 2655 a(\022)1014 2568 y Fe(\023)1045 2577 y Ff(0)1066 2627 y Fp(+)1112 2599 y Fh(H)p 1112 2617 38 2 v 1121 2655 a(\022)1155 2627 y(:)952 2749 y Fm(63)p eop %%Page: 64 6 64 5 bop 0 33 a Fk(I)q(I)47 b(Sym)o(b)q(olic)16 b(Computation)0 141 y Fp(This)e(leads)g(to)g Fh(G)d Fp(=)h Fg(\000)p Fh(z)25 b Fp(mo)q(d)10 b Fh(\022)q(;)21 b Fp(and)14 b Fh(H)g Fp(=)e Fg(\000)p Fh(\022)q(z)r Fp(,)i(th)o(us)h Fh(f)h Fp(=)c(\()p Fg(\000)p Fh(z)r(=\022)q Fp(\))1129 126 y Ff(0)1151 141 y Fg(\000)d Fh(z)16 b Fp(and)1308 108 y Fe(R)1343 141 y Fh(f)g Fp(=)c Fg(\000)p Fh(z)r(=)p Fp(\(tan)7 b Fh(z)r Fp(\))i Fg(\000)h Fh(z)1689 126 y Fl(2)1708 141 y Fh(=)p Fp(2.)50 232 y(3.3.2.)39 b(The)16 b(Rothstein-T)m(rager)f(resultan)o(t)h(W)m(e)f(consider)i Fh(f)h Fp(=)d Fh(f)1117 238 y Fi(p)1147 232 y Fp(+)10 b Fh(f)1209 238 y Fi(s)1237 232 y Fp(+)h Fh(G=D)q Fp(,)k(where)i Fh(D)f Fp(is)g(square-free.)24 b(Again)0 282 y(w)o(e)14 b(consider)734 338 y Fh(R)p Fp(\()p Fh(z)r Fp(\))e(=)g(Res)947 344 y Fi(\022)966 338 y Fp(\()p Fh(G)d Fg(\000)h Fh(z)r(D)1122 320 y Ff(0)1134 338 y Fh(;)d(D)q Fp(\))p Fh(:)50 407 y Fp(The)15 b(function)g Fh(f)20 b Fp(has)15 b(an)g(elemen)o(tary)f(primitiv)o(e)f(if)h(and)h(only)f (if)h Fh(R)p Fp(\()p Fh(z)r Fp(\))e(=)h Fh(\014)r(P)6 b Fp(\()p Fh(z)r Fp(\),)15 b(where)h Fh(\014)h Fg(2)c Fh(K)18 b Fp(and)d Fh(P)k Fg(2)13 b Fh(K)s Fp([)p Fh(z)r Fp(])h(is)0 457 y(a)g(monic)f(p)q (olynomial)d(with)k(constan)o(t)h(co)q(e\016cien)o(ts.)20 b(It)15 b(ma)o(y)d(happ)q(en)j(that)f(the)h(co)q(e\016cien)o(ts)g(of)f Fh(R)p Fp(\()p Fh(z)r Fp(\))g(ha)o(v)o(e)g(a)g(non-zero)0 507 y(deriv)n(ativ)o(e,)f(in)g(whic)o(h)h(case)h Fh(f)k Fp(has)14 b(no)f(elemen)o(tary)h(primitiv)o(e.)50 557 y(The)g(logarithmic)d(part)j(of)g (the)g(primitiv)o(e)e(is)605 600 y Fe(Z)660 629 y Fh(G)p 659 647 36 2 v 659 685 a(D)711 657 y Fp(=)798 618 y Fe(X)755 709 y Fi(\013)p Ff(j)p Fi(R)p Fl(\()p Fi(\013)p Fl(\)=0)908 657 y Fh(\013)6 b Fp(log\(gcd\()p Fh(G)j Fg(\000)h Fh(\013D)1235 640 y Ff(0)1247 657 y Fh(;)d(D)q Fp(\)\))p Fh(:)50 785 y Fp(3.3.3.)39 b(P)o(olynomials)11 b(parts)50 836 y(First)j(w)o(e)g(describ)q(e)i(the)e (metho)q(d)f(for)h(a)f(p)q(olynomial)e(in)i(log\()p Fh(u)p Fp(\()p Fh(z)r Fp(\)\).)18 b(Consider)304 916 y Fh(f)324 922 y Fi(p)355 916 y Fp(=)12 b Fh(a)421 922 y Fi(m)453 916 y Fp(\()p Fh(z)r Fp(\))7 b(log)567 898 y Fi(m)605 916 y Fh(u)p Fp(\()p Fh(z)r Fp(\))i(+)h Fh(a)755 922 y Fi(m)p Ff(\000)p Fl(1)829 916 y Fp(\()p Fh(z)r Fp(\))d(log)943 898 y Fi(m)p Ff(\000)p Fl(1)1024 916 y Fh(u)p Fp(\()p Fh(z)r Fp(\))i(+)h Fg(\001)d(\001)g(\001)g Fp(+)j Fh(a)1273 922 y Fl(1)1291 916 y Fp(\()p Fh(z)r Fp(\))d(log)g Fh(u)p Fp(\()p Fh(z)r Fp(\))j(+)f Fh(a)1562 922 y Fl(0)1581 916 y Fp(\()p Fh(z)r Fp(\))p Fh(:)50 998 y Fp(The)14 b(Liouville)e(principle) i(implies)e(that,)h(if)g Fh(f)776 1004 y Fi(p)810 998 y Fp(admit)f(a)h (primitiv)o(e,)f Fh(f)1174 1004 y Fi(p)1207 998 y Fp(can)i(b)q(e)g(written) 494 1124 y Fh(f)514 1130 y Fi(p)545 1124 y Fp(=)604 1072 y Fi(m)589 1084 y Fe(X)592 1173 y Fi(i)p Fl(=0)656 1124 y Fh(a)678 1130 y Fi(i)692 1124 y Fp(\()p Fh(z)r Fp(\))7 b(log)806 1105 y Fi(i)819 1124 y Fp(\()p Fh(z)r Fp(\))12 b(=)928 1053 y Fe( )981 1072 y Fi(n)961 1084 y Fe(X)964 1173 y Fi(i)p Fl(=0)1028 1124 y Fh(b)1046 1130 y Fi(i)1066 1124 y Fp(log)1120 1105 y Fi(i)1141 1124 y Fh(u)p Fp(\()p Fh(z)r Fp(\))1218 1053 y Fe(!)1251 1061 y Ff(0)1272 1124 y Fp(+)1334 1072 y Fi(k)1313 1084 y Fe(X)1316 1173 y Fi(i)p Fl(=1)1385 1096 y Fh(c)1403 1102 y Fi(i)1417 1096 y Fh(v)1438 1081 y Ff(0)1437 1106 y Fi(i)p 1385 1114 66 2 v 1401 1152 a Fh(v)1421 1158 y Fi(i)0 1241 y Fp(Iden)o(ti\014cation)i(giv)o (es)612 1321 y(0)41 b(=)h Fh(B)781 1304 y Ff(0)779 1332 y Fi(m)p Fl(+1)853 1321 y Fh(;)588 1411 y(A)619 1417 y Fi(i)674 1411 y Fp(=)g Fh(B)781 1394 y Ff(0)779 1421 y Fi(i)802 1411 y Fp(+)10 b(\()p Fh(i)g Fp(+)f(1\))p Fh(B)993 1417 y Fi(i)p Fl(+1)1054 1383 y Fh(u)1078 1368 y Ff(0)p 1054 1402 36 2 v 1060 1440 a Fh(u)1095 1411 y(;)48 b Fp(1)11 b Fg(\024)h Fh(i)26 b Fg(\024)11 b Fh(m:)50 1507 y Fp(Th)o(us)k(eac)o(h)g Fh(B)283 1492 y Ff(0)281 1518 y Fi(i)310 1507 y Fp(can)g(b)q(e)g(iterativ)o(ely)f(computed,)g(and)g (again)g(eac)o(h)h Fh(B)1166 1492 y Ff(0)1164 1518 y Fi(i)1193 1507 y Fp(is)f(in)o(tegrated)h(b)o(y)g(a)f(recursiv)o(e)i(application)e(of)0 1557 y(the)g(algorithm,)d(and)i(is)g(found)h(up)f(to)h(a)f(constan)o(t.)18 b(F)m(urthermore,)13 b(b)o(y)h(iden)o(ti\014cation,)e(the)i(constan)o(t)h(of) e Fh(B)1725 1563 y Fi(i)p Fl(+1)1794 1557 y Fp(is)h(found.)50 1627 y Fa(Example.)155 1722 y Fh(F)j Fp(=)243 1663 y Fe(\022)289 1694 y Fp(3)p 279 1712 42 2 v 279 1750 a(2)p Fh(z)335 1722 y Fp(+)463 1694 y(1)p 381 1712 185 2 v 381 1752 a(log\()p Fh(z)11 b Fp(+)528 1736 y Fl(1)p 528 1743 17 2 v 528 1767 a(2)549 1752 y Fp(\))580 1722 y Fg(\000)e Fp(2)795 1694 y(2)p Fh(z)p 647 1712 337 2 v 647 1752 a Fp(\(2)p Fh(z)i Fp(+)f(1\))d(log)o(\()p Fh(z)12 b Fp(+)946 1736 y Fl(1)p 946 1743 17 2 v 946 1767 a(2)968 1752 y Fp(\))989 1663 y Fe(\023)1026 1722 y Fp(log)1080 1703 y Fl(2)1099 1722 y Fp(\()p Fh(z)r Fp(\))d(+)1245 1694 y(2)e(log)f Fh(z)p 1208 1712 185 2 v 1208 1752 a Fp(log)o(\()p Fh(z)12 b Fp(+)1354 1736 y Fl(1)p 1354 1743 17 2 v 1354 1767 a(2)1376 1752 y Fp(\))1406 1722 y(+)1611 1694 y(2)p 1453 1712 337 2 v 1453 1752 a(\(2)p Fh(z)f Fp(+)f(1\))d(log)o(\()p Fh(z)k Fp(+)1752 1736 y Fl(1)p 1752 1743 17 2 v 1752 1767 a(2)1773 1752 y Fp(\))0 1836 y(W)m(e)j(let)g Fh(\034)j Fp(=)c(log)o(\()p Fh(z)f Fp(+)e(1)p Fh(=)p Fp(2\),)j(and)h Fh(\022)g Fp(=)e(log)7 b Fh(z)r Fp(,)14 b(suc)o(h)h(that)f Fh(F)k Fp(=)12 b Fh(A)1028 1842 y Fl(2)1047 1836 y Fh(\022)1067 1821 y Fl(2)1096 1836 y Fp(+)d Fh(A)1168 1842 y Fl(1)1187 1836 y Fh(\022)i Fp(+)f Fh(A)1290 1842 y Fl(0)1309 1836 y Fh(;)18 b(A)1370 1842 y Fi(i)1397 1836 y Fg(2)11 b Fb(Q)p Fp(\()p Fh(z)r(;)c(\034)e Fp(\).)17 b(A)d(primitiv)o(e)e(has)i(the)0 1885 y(form)e Fh(B)129 1891 y Fl(2)148 1885 y Fh(\022)168 1870 y Fl(3)197 1885 y Fp(+)d Fh(B)269 1891 y Fl(2)288 1885 y Fh(\022)308 1870 y Fl(2)337 1885 y Fp(+)g Fh(B)409 1891 y Fl(1)428 1885 y Fh(\022)i Fp(+)f Fh(B)531 1891 y Fl(0)559 1885 y Fp(+)601 1854 y Fe(P)651 1885 y Fh(c)669 1891 y Fi(i)683 1885 y Fh(v)704 1870 y Ff(0)716 1885 y Fh(i=v)771 1891 y Fi(i)799 1885 y Fp(and:)75 1950 y(\(1\))21 b Fh(B)182 1935 y Ff(0)180 1960 y Fl(3)210 1950 y Fp(=)12 b(0)i(th)o(us)g Fh(B)411 1956 y Fl(3)442 1950 y Fp(=)d Fh(b)503 1956 y Fl(3)533 1950 y Fg(2)p 573 1916 33 2 v 12 w Fb(Q)m Fp(.)75 2000 y(\(2\))21 b Fh(A)180 2006 y Fl(2)210 2000 y Fp(=)12 b Fh(B)287 1984 y Ff(0)285 2010 y Fl(2)313 2000 y Fp(+)d(3)p Fh(B)406 2006 y Fl(3)425 2000 y Fh(\022)445 1984 y Ff(0)458 2000 y Fp(.)18 b(Recursiv)o(ely)c(w)o(e)g(get)839 1966 y Fe(R)874 2000 y Fh(A)905 2006 y Fl(2)935 2000 y Fp(=)e(3)p Fh(\022)q(=)p Fp(2)d(+)g Fh(z)r(=\034)c Fp(,)13 b(this)h(implies)e Fh(b)1442 2006 y Fl(3)1472 2000 y Fp(=)g(1)p Fh(=)p Fp(2)h(and)h Fh(B)1704 2006 y Fl(2)1735 2000 y Fp(=)d Fh(z)r(=\034)j Fp(+)c Fh(b)1912 2006 y Fl(2)1930 2000 y Fp(.)75 2049 y(\(3\))21 b Fh(A)180 2055 y Fl(1)207 2049 y Fg(\000)10 b Fp(2)p Fh(=\034)16 b Fp(=)c Fh(B)402 2034 y Ff(0)400 2060 y Fl(1)428 2049 y Fp(+)e(2)p Fh(b)509 2055 y Fl(2)527 2049 y Fh(\022)547 2034 y Ff(0)559 2049 y Fp(.)18 b(Again)13 b(w)o(e)h(get)840 2016 y Fe(R)875 2049 y Fh(A)906 2055 y Fl(1)934 2049 y Fg(\000)9 b Fp(2)p Fh(=\034)16 b Fp(=)c(0)i(and)f Fh(b)1228 2055 y Fl(2)1258 2049 y Fp(=)f(0,)h Fh(B)1379 2055 y Fl(1)1410 2049 y Fp(=)f Fh(b)1472 2055 y Fl(1)1490 2049 y Fp(.)75 2099 y(\(4\))21 b Fh(A)180 2105 y Fl(0)214 2099 y Fp(=)16 b Fh(B)295 2084 y Ff(0)293 2109 y Fl(0)323 2099 y Fp(+)c Fh(b)385 2105 y Fl(1)403 2099 y Fh(\022)423 2084 y Ff(0)446 2099 y Fp(+)490 2068 y Fe(P)540 2099 y Fh(c)558 2105 y Fi(i)572 2099 y Fh(v)593 2084 y Ff(0)592 2110 y Fi(i)606 2099 y Fh(=v)647 2105 y Fi(i)661 2099 y Fp(.)699 2066 y Fe(R)733 2099 y Fh(A)764 2105 y Fl(0)799 2099 y Fp(=)k(log)6 b(log\()p Fh(z)13 b Fp(+)f(1)p Fh(=)p Fp(2\),)k(th)o(us)h Fh(b)1272 2105 y Fl(1)1306 2099 y Fp(=)f(0,)g Fh(B)1434 2105 y Fl(0)1470 2099 y Fp(is)g(a)h(constan)o(t,)g (and)f Fh(c)1835 2105 y Fl(1)1870 2099 y Fp(=)g(1,)149 2149 y Fh(v)169 2155 y Fl(1)199 2149 y Fp(=)c Fh(\034)5 b Fp(.)0 2213 y(In)14 b(the)g(end,)213 2180 y Fe(R)248 2213 y Fh(F)j Fp(=)12 b(log)o(\()p Fh(z)r Fp(\))442 2198 y Fl(3)461 2213 y Fh(=)p Fp(2)d(+)h Fh(z)f Fp(log)635 2195 y Fl(2)654 2213 y Fp(\()p Fh(z)r Fp(\))p Fh(=)e Fp(log\()p Fh(z)k Fp(+)f(1)p Fh(=)p Fp(2\))e(+)i(log)c(log\()p Fh(z)11 b Fp(+)f(1)p Fh(=)p Fp(2\).)50 2270 y(W)m(e)j(explain)g(the)i(algorithm)c(for)j(a)f(p)q (olynomials)e(in)j(exp)q(onen)o(tial)f(terms:)18 b(let)c Fh(f)1338 2276 y Fi(p)1371 2270 y Fp(b)q(e)248 2351 y Fh(f)268 2357 y Fi(p)299 2351 y Fp(=)e Fh(a)365 2357 y Fi(m)396 2351 y Fp(\()p Fh(z)r Fp(\))7 b(exp)q(\()p Fh(mu)p Fp(\()p Fh(z)r Fp(\)\))j(+)g Fh(a)739 2357 y Fi(m)p Ff(\000)p Fl(1)813 2351 y Fp(\()p Fh(z)r Fp(\))d(exp\(\()p Fh(m)j Fg(\000)g Fp(1\))p Fh(u)p Fp(\()p Fh(z)r Fp(\)\))f(+)h Fg(\001)d(\001)g(\001)h Fp(+)h Fh(a)1358 2357 y Ff(\000)p Fi(p)1403 2351 y Fp(\()p Fh(Z)s Fp(\))e(exp)q([)p Fg(\000)p Fh(pu)p Fp(\()p Fh(z)r Fp(\)])p Fh(:)50 2433 y Fp(The)14 b(Liouville)e(principle)i(implies)e(that)i(the)g(primitiv)o(e)e(is)i(of)f (the)i(form)1215 2401 y Fe(P)1258 2412 y Fi(m)1258 2445 y(i)p Fl(=)p Ff(\000)p Fi(p)1348 2433 y Fh(b)1366 2439 y Fi(i)1379 2433 y Fp(\()p Fh(z)r Fp(\))7 b(exp)q(\()p Fh(iu)p Fp(\()p Fh(z)r Fp(\)\),)14 b(with)848 2519 y Fh(b)866 2502 y Ff(0)866 2529 y Fi(i)888 2519 y Fp(+)c Fh(iu)968 2502 y Ff(0)980 2519 y Fh(b)998 2525 y Fi(i)1023 2519 y Fp(=)i Fh(a)1089 2525 y Fi(i)0 2600 y Fp(whic)o(h)18 b(is)f(called)h(the)g(Risc)o(h)g(di\013eren)o (tial)f(equation.)30 b(This)17 b(equation)h(can)g(b)q(e)g(solv)o(ed)g (without)f(di\016cult)o(y)g(since)i(only)0 2650 y(rational)13 b(solutions)g(are)h(required.)952 2749 y Fm(64)p eop %%Page: 65 7 65 6 bop 1226 33 a Fk(In)o(tro)q(duction)16 b(to)f(sym)o(b)q(olic)h(in)o (tegration)50 141 y Fa(Example.)264 108 y Fe(R)299 141 y Fh(e)318 126 y Ff(\000)p Fi(z)361 114 y Fd(2)379 141 y Fp(.)i(The)d(Risc)o(h)e (di\013eren)o(tial)h(equation)f(is)814 216 y Fh(b)832 199 y Ff(0)843 216 y Fp(\()p Fh(z)r Fp(\))d Fg(\000)f Fp(2)p Fh(z)r(b)p Fp(\()p Fh(z)r Fp(\))j(=)g(1)0 291 y(for)i(whic)o(h)f(a)h(rational)f (solution)g(is)h(to)g(b)q(e)g(found.)k(Because)e(there)f(is)f(no)g(p)q(ole)f (in)h(the)g(co)q(e\016cien)o(ts)i(of)d(the)i(equation)e(and)0 340 y(the)j(leading)f(co)q(e\016cien)o(t)i(is)f(constan)o(t,)g Fh(b)p Fp(\()p Fh(z)r Fp(\))g(has)g(no)g(p)q(ole,)g(and)f(so)h(m)o(ust)f(b)q (e)i(a)e(p)q(olynomial.)21 b(But)16 b(degree)h(constrain)o(ts)0 396 y(sho)o(w)d(this)g(equation)f(has)h(no)g(p)q(olynomial)c(solution.)18 b(Conclusion:)1101 362 y Fe(R)1136 396 y Fh(e)1155 381 y Ff(\000)p Fi(z)1198 368 y Fd(2)1230 396 y Fp(is)c(not)g(elemen)o(tary)m(.)841 483 y Fq(Bibliograp)o(h)o(y)0 558 y Fp([1])20 b(Bronstein)14 b(\(Man)o(uel\).)g({)k(Sym)o(b)q(olic)12 b(in)o(tegration,)g Fh(1993)p Fp(.)h(Bo)q(ok)h(in)f(preparation.)0 608 y([2])20 b(Da)o(v)o(enp)q(ort)13 b(\(J.)h(H.\),)f(Siret)i(\(Y.\),)e(and)h(T)m(ournier) f(\(E.\).)h({)k Fj(Calcul)c(formel)p Fp(.)f({)18 b(P)o(aris,)13 b(Masson,)h Fh(1986)p Fp(.)0 657 y([3])20 b(Geddes)c(\(Keith)h(O.\),)f(Czap)q (or)g(\(Stephen)i(R.\),)d(and)h(Labahn)f(\(George\).)h({)24 b Fj(A)o(lgorithms)16 b(for)g(c)n(omputer)h(algebr)n(a)p Fp(.)e({)65 707 y(Klu)o(w)o(er,)e Fh(1992)p Fp(.)952 2749 y Fm(65)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF