LIST OF FUNCTIONS > with(linalg): The code consists of two main programs - symm and matrices and two auxiliary functions - simple and l_f. The program symm is the main function. The input consists of a complex-valued polynomial f(p) considered as the projective form of homogeneous binary polynomial F(x,y), and the degree n=deg(F). The program computes the invariants J and K in reduced form, determines the dimension of the symmetry group, and, in the case of a finite symmetry group, applies the Maple command solve to solve the two polynomial symmetry equations (3,4) to find explicit form of symmetries. The output of symm consists of the projective index of the form and the explicit formulae for its discrete projective symmetries. The program also notifies the user if the symmetry group is not discrete, or is in the maximal discrete symmetry class. > symm:=proc(form,n) > global tr,error; > local Q,Qp,Qpp,Qppp,Qpppp,H,T,V,U,J,K,j,k, Eq1,Eq2,i,eqtr,ans; > tr:='tr': > Q:=form(p); > Qp:=diff(Q,p); > Qpp:=diff(Qp,p); > Qppp:=diff(Qpp,p); > Qpppp:=diff(Qppp,p); > H:=n*(n-1)*(Q*Qpp-(n-1)/n*Qp^2); > if H=0 then > ans:=Hessian is zero: two-dimensional symmetry group > else > T:=-n^2*(n-1)*(Q^2*Qppp-3*(n-2)/n*Q*Qp*Qpp+2*(n-1)*(n-2)/n^2 *Qp^3); > 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; > U:=n^3*(n-1)*V-3*(n-2)/(n-1)*H^2; > J:=simple(T^2/H^3);K:=simple(U/H^2); > j:=subs(p=P,J);k:=subs(p=P,K); > Eq1:=simplify(numer(J)*denom(j)-numer(j)*denom(J)); > Eq2:=simplify(numer(K)*denom(k)-numer(k)*denom(K)); > if Eq1=0 then > ans:=Form has a one-dimensional symmetry group; > else > if Eq2=0 then > print ( Form has the maximal possible discrete symmetry group); > eqtr:= [solve(Eq1,P)]; > tr:=map(radsimp,map(allvalues,eqtr)); > else > eqtr:=[solve({Eq1,Eq2},P)]; > tr:= []; > for i from 1 to nops(eqtr) do > tr:=map(radsimp,[op(tr),allvalues(rhs(eqtr[i][1]))]); > od > fi; > print(The number of the projective symmetries=nops(tr)); > ans:=map(l_f,tr); > if error=1 then > print(ERROR: Some of the transformations are not > linear-fractional) > else > if error=2 then > print(WARNING: Some of the transformations are not written in the form polynomial over polynomial) > fi; > fi; > fi; > fi; > ans > end: > The program matrices determines the matrix symmetry corresponding to a given (list of) projective symmetries. As discussed in the text, this only requires determining an overall scalar multiple, which can be found by substituting the projective symmetry into the form. The output consists of each projective symmetry, the scalar factor $\mu$, and the resulting matrix symmetry. > matrices:=proc(form,n,L::list) > local Q,ks,ksi,i,Sf,M; > ksi:='ksi'; > for i from 1 to nops(L) do > Sf:=simplify(denom(L[i])^n*form(L[i])); > ks:=quo(Sf,form(p),p); > ksi:=simplify(ks^(1/n),radical,symbolic); > M[i]:=matrix(2,2,[coeff(numer(L[i]),p)/ksi, > coeff(numer(L[i]),p,0)/ksi,coeff(denom(L[i]),p)/ksi, > coeff(denom(L[i]),p,0)/ksi]); > print(L[i], mu=ksi, map(simplify,M[i])); > od; > end: > The auxiliary function l_f uses polynomial division to reduce rational expressions to linear fractional form (when possible). > l_f:=proc(x) > local A,B,C,S,de,nu,r,R; > global error;error:='error'; > nu:=numer(x); > de:=denom(x); > if type(nu,polynom(anything,p)) and type(de,polynom(anything,p)) > then > if degree(nu,p)+1=degree(de,p) then > A:=quo(de,nu,p,'B'); > S:=1/A;R:=0 > else > A:=quo(nu,de,p,'B'); > if B=0 then S:=A; > R:=0; > else > C:=quo(de,B,p,'r');R:=simple(r); > S:=simplify(A+1/C) > fi; > fi; > if R=0 then > collect(S,p) > else > error:=1;x > fi; > else > error:=2;x fi; > end: The auxiliary function simple helps to simplify rational expressions by manipulating the numerator and denominator separately. The simplified rational expression is returned. > simple:=proc(x) > local nu,de,num,den; > nu:=numer(x); > de:=denom(x); > num:=(simplify((nu,radical,symbolic))); > den:=(simplify((de,radical,symbolic))); > simplify(num/den); > end: > > Example: > f:=p->p^3+1; 3 f := p -> p + 1 > symm(f,3); Form has the maximal possible discrete symmetry gr\ oup The number of the projective symmetries = 6 -1/2 + 1/2 I sqrt(3) -1/2 - 1/2 I sqrt(3) [p, 1/p, --------------------, --------------------, p p (-1/2 + 1/2 I sqrt(3)) p, (-1/2 - 1/2 I sqrt(3)) p] > matrices(f,3,%); [1 0] p, mu = 1, [ ] [0 1] [0 1] 1/p, mu = 1, [ ] [1 0] -1/2 + 1/2 I sqrt(3) [0 -1/2 + 1/2 I sqrt(3)] --------------------, mu = 2, [ ] p [1 0 ] -1/2 - 1/2 I sqrt(3) [0 -1/2 - 1/2 I sqrt(3)] --------------------, mu = 2, [ ] p [1 0 ] [-1/2 + 1/2 I sqrt(3) 0] (-1/2 + 1/2 I sqrt(3)) p, mu = 2, [ ] [ 0 1] [-1/2 - 1/2 I sqrt(3) 0] (-1/2 - 1/2 I sqrt(3)) p, mu = 2, [ ] [ 0 1] > f:=p->p^2; 2 f := p -> p > symm(f,3); Form has a one-dimensional symmetry group >