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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Set home directory" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "currentdir(\"C://Documents a
nd Settings/mengland/My Documents/Maple/DM-35\"):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 28 "We work with the (3,5)-curve" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "n:=3: s:=5: genus:=4;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&genusG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Suppose w
e want to find the coefficients of an equation in P functions." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "rel:= 0 = P(3,3,4,4) + c[1]*P(2,3) \+
+ c[2]*P(3,4)*lambda[4] + c[3]*P(3,3)*P(4,4)+c[4]*P(3,4)^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$relG/\"\"!,,-%\"PG6&\"\"$F+\"\"%F,\"\"\"*
&&%\"cG6#F-F--F)6$\"\"#F+F-F-*(&F06#F4F--F)6$F+F,F-&%'lambdaG6#F,F-F-*
(&F06#F+F--F)6$F+F+F--F)6$F,F,F-F-*&&F0F<F-)F8F4F-F-" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 73 "We substitute the P-functions for sigma functio
ns using their definition." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "read(
\"p5subs.txt\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "polnum:
=numer(factor(subs(psubs,rhs(rel)))): nops(%); op(1,%%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%d
iffG6%-%#siG6&%#u1G%#u2G%#u3G%#u4G-%\"$G6$F-\"\"#-F06$F.F2\"\"\")F(\"
\"$F5!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Notice that each of
 the terms is a product of 4-sigma derivatives." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We will substitute sigma \+
for its series and multiply out the product.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We read in the sigma exp
ansion.  We only need the first two parts as the polynomial only invol
ves lambda[4]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "read(\"SW.txt\"):
 read(\"C11.txt\"): sigma:=unapply(SW+C11, v[1],v[2],v[3],v[4]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6&%$v_1G%$v_2G%$v_3G%$v_4G6
\"6$%)operatorG%&arrowGF+,8*&#\"\"\"\"$[%F2*$)9'\"\")F2F2F2*&#F2F7F2*&
)9&\"\"#F2)F6\"\"%F2F2!\"\"*&#F2F?F2*$)F<F?F2F2F@*$)9%F=F2F2*(F<F2)F6F
=F2FGF2F2*&F6F29$F2F@*&#F2\"%SAF2*(&%'lambdaG6#F?F2)F6\"\"*F2F<F2F2F2*
&#F2\"#SF2*(FPF2)F6\"\"&F2)F<\"\"$F2F2F@*&#F2\"#?F2*(FPF2F6F2)F<FZF2F2
F@*&#F2\"$S\"F2*(FPF2)F6\"\"(F2FGF2F2F2**FPF2F6F2F<F2FFF2F2F+F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "We could just sub in, expand the p
roduct, and discard the invalud terms with higher lambda..." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "ONE:=algsubs(lambda[4]^2=0,expand(eval(su
bs(si=sigma,polnum)))): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT -1 81 "Then setting each coefficient to zero gives linear equati
ons we can easily solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
100 "map(coeffs,map(coeffs,map(coeffs,map(coeffs, \{coeffs(ONE,lambda[
4])\},u1),u2),u3),u4): sol:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$solG<&/&%\"cG6#\"\"\"F*/&F(6#\"\"#!\"#/&F(6#\"\"$F//&F(6#\"\"%!
\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 20 "to give the
 relation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(sol,rel);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,,-%\"PG6&\"\"$F)\"\"%F*\"\"\"-F
'6$\"\"#F)F+*(F.F+-F'6$F)F*F+&%'lambdaG6#F*F+!\"\"*(F.F+-F'6$F)F)F+-F'
6$F*F*F+F5*&F*F+)F0F.F+F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "But
 in general it is far more efficient to use the code Expand_Product wh
ich takes advantage of the weight properties." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 120 "We use the code on each product in turn.  The second \+
argument is the weight at which we discard terms, 4 in this case.  " }
}{PARA 0 "" 0 "" {TEXT -1 77 "The third argument is the number of seri
es to multiply, again 4 in this case." }}{PARA 0 "" 0 "" {TEXT -1 66 "
The fourth argument is an optional set of further simplifications." }}
{PARA 0 "" 0 "" {TEXT -1 84 "The fifth arguement is also optional, for
 use with the addition formula operators.  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "read(\"Expand_Product.txt\"):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 99 "TWO:=0: for i from 1 to nops(polnum) do TWO:=T
WO + Expand_Product( op(i,polnum), 4,4, [], no): od: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "map(coeffs,map(coeffs,map(coeffs,m
ap(coeffs, \{coeffs(TWO,lambda[4])\},u1),u2),u3),u4): sol:=solve(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<&/&%\"cG6#\"\"\"F*/&F(6#\"\"
#!\"#/&F(6#\"\"$F//&F(6#\"\"%!\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "subs(sol,rel);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"
\"!,,-%\"PG6&\"\"$F)\"\"%F*\"\"\"-F'6$\"\"#F)F+*(F.F+-F'6$F)F*F+&%'lam
bdaG6#F*F+!\"\"*(F.F+-F'6$F)F)F+-F'6$F*F*F+F5*&F*F+)F0F.F+F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 141 "In this toy exampl
e the efficiency is not highlighted.  But with larger polynomials at h
igher weight the second method is far more efficient." }}}}{MARK "16 0
 1" 57 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }