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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "LIST OF FUNCTIONS" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 41 "The code consists of two main programs - \+
" }{TEXT 276 4 "symm" }{TEXT -1 8 "  and   " }{TEXT 277 9 "matrices " 
}{TEXT -1 33 "  and two auxiliary  functions - " }{TEXT 278 6 "simple
" }{TEXT -1 7 "  and  " }{TEXT 279 6 " l_f.\n" }{TEXT -1 13 "The progr
am  " }{TEXT 280 4 "symm" }{TEXT -1 78 "  is  the main function. \nThe
 input consists of a complex-valued polynomial   " }{TEXT 281 4 "f(p)
" }{TEXT -1 69 "  considered as the\nprojective form of homogeneous bi
nary polynomial " }{TEXT 282 6 "F(x,y)" }{TEXT -1 17 ", and\nthe degre
e " }{TEXT 283 9 " n=deg(F)" }{TEXT -1 40 ".   The program computes th
e invariants " }{TEXT 284 4 " J  " }{TEXT -1 5 "and  " }{TEXT 285 1 "K
" }{TEXT -1 140 "  in reduced form, determines the dimension of the sy
mmetry group, and, in the case of a finite symmetry group, applies the
\n Maple command  " }{TEXT 286 5 "solve" }{TEXT -1 93 "  to solve the \+
two polynomial symmetry equations  (3,4) to find\nexplicit form of sym
metries. " }}{PARA 0 "" 0 "" {TEXT -1 15 "The output of  " }{TEXT 287 
4 "symm" }{TEXT -1 237 "  consists of the projective index of the form
 and the explicit formulae for its\ndiscrete   projective symmetries. \+
 The program also notifies the user if the\nsymmetry group is not disc
rete,  or is in the maximal discrete symmetry class. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1487 "symm:=proc(form,n)\nglobal tr,err
or;\nlocal Q,Qp,Qpp,Qppp,Qpppp,H,T,V,U,J,K,j,k, Eq1,Eq2,i,eqtr,ans;\n \+
tr:='tr':\n Q:=form(p);\n Qp:=diff(Q,p);\n Qpp:=diff(Qp,p);\n Qppp:=di
ff(Qpp,p);\n Qpppp:=diff(Qppp,p);\n H:=n*(n-1)*(Q*Qpp-(n-1)/n*Qp^2);\n
 if H=0 then\n  ans:=`Hessian is zero: two-dimensional symmetry group`
\n else\n  T:=-n^2*(n-1)*(Q^2*Qppp-3*(n-2)/n*Q*Qp*Qpp+2*(n-1)*(n-2)/n^
2      *Qp^3);\n V:=Q^3*Qpppp-4*(n-3)/n*Q^2*Qp*Qppp+6*(n-2)*(n-3)/n^2*
Q*Qp^2  *Qpp-3*(n-1)*(n-2)*(n-3)/n^3*Qp^4;\n U:=n^3*(n-1)*V-3*(n-2)/(n
-1)*H^2;\n J:=simple(T^2/H^3);K:=simple(U/H^2);\n j:=subs(p=P,J);k:=su
bs(p=P,K);\n Eq1:=simplify(numer(J)*denom(j)-numer(j)*denom(J));\n Eq2
:=simplify(numer(K)*denom(k)-numer(k)*denom(K));\n if Eq1=0 then\n   a
ns:=`Form has a one-dimensional symmetry group`;\n else\n   if Eq2=0  \+
then\n    print (` Form has the maximal possible discrete symmetry    \+
          group`);\n eqtr:= [solve(Eq1,P)];\n        tr:=map(radsimp,m
ap(allvalues,eqtr));\n     else\n        eqtr:=[solve(\{Eq1,Eq2\},P)];
\n        tr:= [];\n        for i from 1 to nops(eqtr) do\n         tr
:=map(radsimp,[op(tr),allvalues(rhs(eqtr[i][1]))]);\n        od\n     \+
 fi;\n    print(`The number of the projective symmetries`=nops(tr));\n
    ans:=map(l_f,tr);\n    if error=1 then\n     print(`ERROR: Some of
 the transformations are not\n     linear-fractional`)\n    else\n    \+
 if error=2 then\n      print(`WARNING: Some of the transformations ar
e not written    in the form polynomial over polynomial`)\n     fi;\n \+
   fi;\n   fi;\n fi;\n ans\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
272 0 "" }{TEXT -1 13 "The program  " }{TEXT 289 8 "matrices" }{TEXT 
-1 361 "  determines the matrix\nsymmetry corresponding to a given (li
st of) projective symmetries.  As discussed in the text, this only\nre
quires determining an overall scalar multiple, which can be found by s
ubstituting the projective\nsymmetry into the form.  The output consis
ts of each projective symmetry, the scalar factor $\\mu $,\nand the re
sulting matrix symmetry." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
399 "matrices:=proc(form,n,L::list)\n local Q,ks,ksi,i,Sf,M;\n ksi:='k
si';\n for i from 1 to nops(L) do\n  Sf:=simplify(denom(L[i])^n*form(L
[i]));\n  ks:=quo(Sf,form(p),p);\n  ksi:=simplify(ks^(1/n),radical,sym
bolic);\n  M[i]:=matrix(2,2,[coeff(numer(L[i]),p)/ksi,\n  coeff(numer(
L[i]),p,0)/ksi,coeff(denom(L[i]),p)/ksi,\n  coeff(denom(L[i]),p,0)/ksi
]);\n  print(L[i],    mu=ksi,   map(simplify,M[i])); \n od;\nend:\n" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The auxiliary function  " }
{TEXT 288 5 "l_f  " }{TEXT -1 100 "uses polynomial division  to\nreduc
e rational expressions to linear fractional form (when possible). " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 544 "l_f:=proc(x)\n local A,B,C
,S,de,nu,r,R;\n global error;error:='error';\n nu:=numer(x);  \n de:=d
enom(x);\n if type(nu,polynom(anything,p)) and     type(de,polynom(any
thing,p))\n then\n  if degree(nu,p)+1=degree(de,p) then\n   A:=quo(de,
nu,p,'B');\n   S:=1/A;R:=0\n  else \n   A:=quo(nu,de,p,'B');\n   if B=
0 then S:=A;\n     R:=0;   \n   else\n     C:=quo(de,B,p,'r');R:=simpl
e(r);\n     S:=simplify(A+1/C) \n   fi;\n  fi;\n  if R=0 then   \n    \+
collect(S,p)\n  else\n    error:=1;x\n  fi;\n else\n  error:=2;x      \+
                                             fi;\nend:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "The auxiliary function " }{TEXT 290 7 "si
mple " }{TEXT -1 147 " helps to simplify rational\n expressions by man
ipulating the numerator and denominator separately.\nThe simplified ra
tional expression is returned.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 170 "simple:=proc(x)\nlocal nu,de,num,den;\nnu:=numer(x);
\nde:=denom(x);\nnum:=(simplify((nu,radical,symbolic)));\nden:=(simpli
fy((de,radical,symbolic)));\nsimplify(num/den);\nend:\n\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "f:=p->p^3+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f
GR6#%\"pG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"\"\"\"F2F2F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "symm(f,3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%^o~Form~has~the~maximal~possible~discrete~symm
etry~~~~~~~~~~~~~~groupG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%HThe~num
ber~of~the~projective~symmetriesG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7(%\"pG*&\"\"\"F&F$!\"\"*&,&#!\"\"\"\"#\"\"\"*&%\"IGF--%%sqrtG6#
\"\"$F&#F-F,F&F$F'*&,&F*F-F.F*F&F$F'*&F)F-F$F-*&F6F-F$F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "matrices(f,3,%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%%\"pG/%#muG\"\"\"-%'matrixG6#7$7$F&\"\"!7$F,F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%*&\"\"\"F$%\"pG!\"\"/%#muG\"\"\"-%'mat
rixG6#7$7$\"\"!F)7$F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"
\"\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"#F(F'F/%\"pG!\"\"/%#muGF'-%'m
atrixG6#7$7$\"\"!F$7$F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"
\"\"\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"F%F/%\"pG!\"\"/%#muGF'-%'ma
trixG6#7$7$\"\"!F$7$F(F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"
\"\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"#F(F'F(%\"pGF(/%#muGF'-%'matr
ixG6#7$7$F$\"\"!7$F9F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"\"
\"#\"\"\"*&%\"IGF(-%%sqrtG6#\"\"$\"\"\"F%F(%\"pGF(/%#muGF'-%'matrixG6#
7$7$F$\"\"!7$F8F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=p->
p^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"pG6\"6$%)operatorG
%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "symm(f,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%JForm~
has~a~one-dimensional~symmetry~groupG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "2 0 10" 4 }{VIEWOPTS 1 1 0 1 1 1803 }