{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 22 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet It em" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "AbelMap" }{TEXT 30 63 " - compute \+ the Abel map between two points on a Riemann surface" }}{PARA 4 "" 0 " usage" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 39 " \+ AbelMap(F, x, y, P, P_0, accuracy)" }}{PARA 0 "" 0 "" {TEXT -1 29 " AbelMap(F, x, y, P, P_0)" }}{PARA 4 "" 0 "" {TEXT -1 10 "Paramet ers" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 11 "F - " } {TEXT -1 26 "irreducible polynomial in " }{TEXT 22 1 "x" }{TEXT -1 5 " and " }{TEXT 22 1 "y" }{TEXT -1 35 " specifying a Riemann surface by \+ F(" }{TEXT 22 1 "x" }{TEXT -1 1 "," }{TEXT 22 1 "y" }{TEXT -1 5 ") = 0 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 11 "x - " } {TEXT -1 8 "variable" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 11 "y - " }{TEXT -1 8 "variable" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 29 "P - an x,y pair (or a " }{TEXT -1 38 "Puis eux representation in a parameter " }{TEXT 22 2 "t)" }{TEXT -1 51 " o f a point on the Riemann surface specified by F(" }{TEXT 22 1 "x" } {TEXT -1 1 "," }{TEXT 22 1 "y" }{TEXT -1 3 ")=0" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 11 "P_0 - " }{TEXT -1 9 "same as P" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 11 "accuracy - " } {TEXT -1 60 "number of desired accurate decimal digits, default is Dig its" }}{SECT 0 {PARA 4 "" 0 "info" {TEXT -1 11 "Description" }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 35 7 "Abelmap" }{TEXT -1 50 " comm and computes the Abel map between two points " }{XPPEDIT 18 0 "P" "6#% \"PG" }{TEXT -1 5 " and " }{TEXT 22 3 "P_0" }{TEXT -1 22 " on a Rieman n surface " }{TEXT 22 1 "R" }{TEXT -1 10 " of genus " }{TEXT 22 1 "g" }{TEXT -1 165 ", that is a g-tuple of complex numbers. The j-th elemen t of the Abel map is the integal of the j-th normalized holomorphic di fferential integrated along a path from " }{TEXT 22 1 "P" }{TEXT -1 4 " to " }{TEXT 22 3 "P_0" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 34 "The Riemann surface is entered as " }{TEXT 22 1 "F" }{TEXT -1 18 " ; an irreducible, " }{HYPERLNK 17 "square-free" 2 "sqrfree" "" }{TEXT -1 15 " polynomial in " }{TEXT 22 1 "x" }{TEXT -1 5 " and " }{TEXT 22 1 "y" }{TEXT -1 60 ". Floating point numbers are not allowed as coeffi cients of " }{TEXT 22 1 "F" }{TEXT -1 60 ". Algebraic numbers are allo wed. Curves of arbitrary finite " }{HYPERLNK 17 "genus" 2 "algcurves[g enus]" "" }{TEXT -1 42 " with arbitrary singularities are allowed." }} {PARA 15 "" 0 "" {TEXT -1 11 "The points " }{TEXT 22 1 "P" }{TEXT -1 5 " and " }{TEXT 22 3 "P_0" }{TEXT -1 26 " are entered either as an " }{TEXT 35 5 "x, y " }{TEXT -1 1 "p" }{TEXT -1 6 "air as" }{TEXT 256 18 " [x = a, y = b] " }{TEXT -1 33 "or as a Puiseux representation: " }{TEXT -1 1 " " }{TEXT 35 40 "[x = a+b*t^r, y = (Laurent series in \+ t)]" }{TEXT -1 8 ", where " }{TEXT 22 1 "a" }{TEXT -1 5 " and " } {TEXT 22 1 "b" }{TEXT -1 20 " are constants, and " }{TEXT 22 1 "r" } {TEXT -1 19 " is an integer. If " }{TEXT 22 1 "r" }{TEXT -1 49 " < 0, \+ that is, if entering one of the points for " }{TEXT 22 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 7 ", th en " }{TEXT 22 1 "a" }{TEXT -1 324 " = 0. The points P and P_0 may be \+ entered in different forms, that is, P may be input as an x,y pair whi le P_0 is input as a Puiseux representation. Note that for regular poi nts on the Riemann surface the x,y pair representation suffices. For d iscriminant points and points at infinity a Puiseux representation is \+ necessary." }}{PARA 15 "" 0 "" {TEXT -1 47 "The differentials are norm alized such that the " }{TEXT 22 4 "j-th" }{TEXT -1 36 " differential \+ integrated around the " }{TEXT 22 4 "k-th" }{TEXT -1 20 " cycle, as gi ven by " }{HYPERLNK 17 "algcurves[homology]" 2 "algcurves[homology]" " " }{TEXT -1 5 ", is " }{HYPERLNK 17 "Kronecker delta" 2 "Physics/kd_" "" }{TEXT -1 2 " (" }{TEXT 22 1 "j" }{TEXT -1 2 ", " }{TEXT 22 1 "k" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 107 "Note: The Abel map wi ll almost always be computed along with other objects associated with \+ some polynomial " }{TEXT 22 1 "F" }{TEXT -1 183 ", such as the Riemann matrix. It is imperitive that the order of the differential be the sa me for each of the objects, and at each stage of the calculation. As n o order is imposed by " }{HYPERLNK 17 "algcurves[differentials]" 2 "al gcurves[differentials]" "" }{TEXT -1 23 ", make sure to compute " } {TEXT 35 7 "AbelMap" }{TEXT -1 19 " and, for instance " }{HYPERLNK 17 "algcurves[periodmatrix]" 2 "algcurves[periodmatrix]" "" }{TEXT -1 41 ", without a restart (or quit) in between." }}}{SECT 0 {PARA 4 "" 0 "e xamples" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(algcurves, AbelMap, genus, puiseux);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%(AbelMapG%&genusG%(puiseuxG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f := y^2-(x^2-1)*(x^2-4)*(x^2-9)*(x^2-16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$%\"yG\"\"#\"\"\"**,&*$%\"xGF (F)!\"\"F)F),&F,F)!\"%F)F),&F,F)!\"*F)F),&F,F)!#;F)F)F." }}}{PARA 0 " " 0 "" {TEXT -1 31 "Give a look first at the genus " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "genus(f, x, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pui seux(f, x=1, y, 0, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#7$/%\"xG,& *$%\"tG\"\"#!$?(\"\"\"F,/%\"yG,$F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "puiseux(f, x=4, y, 0, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#7$/%\"xG,&*$%\"tG\"\"#\"&!35\"\"%\"\"\"/%\"yG,$F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Compute the Abel map for this curve " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P_0, P := op(%%), op(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%$P_0G%\"PG6$7$/%\"xG,&*$%\"tG\" \"#!$?(\"\"\"F0/%\"yG,$F-F/7$/F*,&F,\"&!35\"\"%F0/F2,$F-F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A := AbelMap(f, x, y, P, P_0, 7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%^$$!+&RKn3&!#5$!+M$=eR\"!\"* ^$$\"+O_Ye^F)$\"+f.CLPF)^$$\"*Zv*pr!#6$!+F3F&e$F)" }}}}{SECT 0 {PARA 4 "" 0 "seealso" {TEXT -1 9 "See Also " }}{PARA 0 "" 0 "" {HYPERLNK 17 "algcurves[puiseux]" 2 "algcurves[puiseux]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "algcurves[differentials]" 2 "algcurves[differentials]" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "algcurves[genus]" 2 "algcurves[genus ]" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}}{MARK "12 9 0 0" 30 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }