Invariants (Section 7)
The 4 equations
> | e1:=subs(y=Y1, Ker1); e2:=subs(y=Y2, Ker1); e3:=subs(y=Y1, R); e4:=subs(y=Y2, R); |
Eliminate Q1(x) between e1 and e2...
> | eqx1:=collect(Y2*e1-Y1*e2,[Q,Q1],factor); |
and then between e3 and e4
> | eqx2:=collect(e3-e4,[Q,Q1],factor); |
Eliminate x between eqx1 and eqx2: this will give the first invariant
> | res:=factor(resultant(numer(eqx1), numer(eqx2),x)); |
> | nops(res); |
> | e5:=collect(-op(1,op(7,res)),Q,factor); |
> | e6:=map(Z->factor(Z/Y1^2/Y2^2),e5); |
> | convert(op(3,e6),parfrac,Y1); |
> | convert(%,parfrac,Y2); |
> | e7:=subsop(3=+1/Y1-t/Y1^2+t/Y2^2-1/Y2,e6); |
> | factor(e6/e7); |
The first invariant
> | I1:=t*y*Q(0,y)*q-1/y+t/y^2; |
> | factor((subs(y=Y1,I1)-subs(y=Y2,I1))/e6); |
Expression of Q(0,Y_i) in terms of the invariant
> | Qysol:=factor(isolate(I1-II,Q(0,y))); |
> | e8:=subs(subs(y=Y1,Qysol), subs(y=Y2,Qysol),subs(isolate(subs(Q1(x)=Q1,e3),Q1),subs(Q1(x)=Q1,e4))); |
> | e9:=subs(subs(y=Y1,Qysol), subs(y=Y2,Qysol),subs(isolate(subs(Q1(x)=Q1,e1),Q1),subs(Q1(x)=Q1,e3))); |
> | res:=factor(resultant(numer(e8),numer(e9),x)); |
> | nops(res); |
> | seq(op(i,res),i=1..6);; |
> | e10:=map(factor,collect(op(1,op(7,res)),[Y1,Y2],distributed)); |
Version of the paper
> | (nu-1)*t^2*(y1^2+y2^2-(q-2)*y1*y2)+t*y1*y2*(y1+y2)*(t*q*nu*II+(q-2)*(nu-1)) +y1^2*y2^2*(q*(1-2*nu)*t*II+t^3*q^2*nu-(q-1)*(nu-1)); |
> | factor(subs(y1=Y1,y2=Y2,%)+e10); |
> | eq14:=collect(numer(subs(Y1=1/Z1,Y2=1/Z2,e10)),[Z1,Z2],factor); |
> | collect(subs(Z1=V1+a,Z2=V2+a,eq14),[V1,V2],factor); |
> | asol:=factor(solve(subs(V2=0,coeff(%,V1,1)),a)); |
> | eq15:=collect(subs(Z1=V1+asol,Z2=V2+asol,-eq14/t^2/(nu-1)),[V1,V2],factor); |
Le coefficient de V1 V2 est bien -(q-2).
We will have to take the square root of
> | CC:=-collect(subs(nu=1+beta,factor(op(4,eq15)*t^2*(nu-1)^2*(4-q)/q)),[II,t],factor); |
> |