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);

e1 := -t*Y1^2*(-q+q*x*nu*t-nu+1)*Q(0,Y1)/(-1+x*nu*t)-Y1*(nu+x*nu*t-1)*Q1(x)/nu+(nu-1)*t*Y1/(-1+x*nu*t)/x-x*t+1-t/Y1

e2 := -t*Y2^2*(-q+q*x*nu*t-nu+1)*Q(0,Y2)/(-1+x*nu*t)-Y2*(nu+x*nu*t-1)*Q1(x)/nu+(nu-1)*t*Y2/(-1+x*nu*t)/x-x*t+1-t/Y2

e3 := -(nu-1)*t*Y1/(1-x*nu*t)/x*Q(0,Y1)-t*Q1(x)+1-t/Y1-x*t

e4 := -(nu-1)*t*Y2/(1-x*nu*t)/x*Q(0,Y2)-t*Q1(x)+1-t/Y2-x*t

Eliminate Q1(x) between e1 and e2...

>    eqx1:=collect(Y2*e1-Y1*e2,[Q,Q1],factor);

eqx1 := -Y2*t*Y1^2*(-q+q*x*nu*t-nu+1)/(-1+x*nu*t)*Q(0,Y1)+Y1*t*Y2^2*(-q+q*x*nu*t-nu+1)/(-1+x*nu*t)*Q(0,Y2)+(-Y2+Y1)*(Y1*x*t*Y2+t*Y1-Y1*Y2+t*Y2)/Y1/Y2

and then between e3 and  e4

>    eqx2:=collect(e3-e4,[Q,Q1],factor);

eqx2 := (nu-1)*t*Y1/(-1+x*nu*t)/x*Q(0,Y1)-(nu-1)*t*Y2/(-1+x*nu*t)/x*Q(0,Y2)+t*(-Y2+Y1)/Y1/Y2

Eliminate x between eqx1 and eqx2: this will give the first invariant

>    res:=factor(resultant(numer(eqx1), numer(eqx2),x));

res := t^5*Y1*Y2*nu^3*(-Y2+Y1)*(Y1*Q(0,Y1)-Y2*Q(0,Y2))*(Y2^2*t*Y1^3*Q(0,Y1)*q-t*Y1^2+Y1^2*Y2-Y1^2*t*Y2^3*Q(0,Y2)*q-Y2^2*Y1+t*Y2^2)^2*(nu-1)

>    nops(res);

8

>    e5:=collect(-op(1,op(7,res)),Q,factor);

e5 := -Y2^2*t*Y1^3*Q(0,Y1)*q+Y1^2*t*Y2^3*Q(0,Y2)*q-(-Y2+Y1)*(Y1*Y2-t*Y1-t*Y2)

>    e6:=map(Z->factor(Z/Y1^2/Y2^2),e5);

e6 := -t*Y1*Q(0,Y1)*q+t*Y2*Q(0,Y2)*q-(-Y2+Y1)*(Y1*Y2-t*Y1-t*Y2)/Y1^2/Y2^2

>    convert(op(3,e6),parfrac,Y1);

-(Y2-t)/Y2^2-t/Y1^2+1/Y1

>    convert(%,parfrac,Y2);

(Y1-t)/Y1^2+t/Y2^2-1/Y2

>    e7:=subsop(3=+1/Y1-t/Y1^2+t/Y2^2-1/Y2,e6);

e7 := -t*Y1*Q(0,Y1)*q+t*Y2*Q(0,Y2)*q+1/Y1-t/Y1^2+t/Y2^2-1/Y2

>    factor(e6/e7);

1

The first invariant

>    I1:=t*y*Q(0,y)*q-1/y+t/y^2;

I1 := t*y*Q(0,y)*q-1/y+t/y^2

>    factor((subs(y=Y1,I1)-subs(y=Y2,I1))/e6);

-1

Expression of Q(0,Y_i) in terms of the invariant

>    Qysol:=factor(isolate(I1-II,Q(0,y)));

Qysol := Q(0,y) = (y-t+II*y^2)/y^3/t/q

>    e8:=subs(subs(y=Y1,Qysol), subs(y=Y2,Qysol),subs(isolate(subs(Q1(x)=Q1,e3),Q1),subs(Q1(x)=Q1,e4)));

e8 := -(nu-1)/Y2^2/(1-x*nu*t)/x*(Y2-t+II*Y2^2)/q+(nu-1)/Y1^2/(1-x*nu*t)/x*(Y1-t+II*Y1^2)/q+t/Y1-t/Y2

>    e9:=subs(subs(y=Y1,Qysol), subs(y=Y2,Qysol),subs(isolate(subs(Q1(x)=Q1,e1),Q1),subs(Q1(x)=Q1,e3)));

e9 := -(nu-1)/Y1^2/(1-x*nu*t)/x*(Y1-t+II*Y1^2)/q+t*(1/Y1*(-q+q*x*nu*t-nu+1)*(Y1-t+II*Y1^2)/q/(-1+x*nu*t)-(nu-1)*t*Y1/(-1+x*nu*t)/x+x*t-1+t/Y1)/Y1/(nu+x*nu*t-1)*nu+1-t/Y1-x*t

>    res:=factor(resultant(numer(e8),numer(e9),x));

res := (-Y2+Y1)^4*Y1^4*t^6*q^2*nu^4*(nu-1)^2*(t*Y2*q*Y1^2+t*Y2^2*q*Y1-Y1^2*Y2^2*q+Y2^2*t^2-Y1^2*t^2*Y2*q*nu*II-Y1^2*t*Y2^2*q*II+2*Y1^2*t*Y2^2*q*nu*II-Y1*t^2*Y2^2*q*nu*II+Y1*t^2*Y2*q*nu-Y1*t^2*Y2*q-Y1*t...
res := (-Y2+Y1)^4*Y1^4*t^6*q^2*nu^4*(nu-1)^2*(t*Y2*q*Y1^2+t*Y2^2*q*Y1-Y1^2*Y2^2*q+Y2^2*t^2-Y1^2*t^2*Y2*q*nu*II-Y1^2*t*Y2^2*q*II+2*Y1^2*t*Y2^2*q*nu*II-Y1*t^2*Y2^2*q*nu*II+Y1*t^2*Y2*q*nu-Y1*t^2*Y2*q-Y1*t...
res := (-Y2+Y1)^4*Y1^4*t^6*q^2*nu^4*(nu-1)^2*(t*Y2*q*Y1^2+t*Y2^2*q*Y1-Y1^2*Y2^2*q+Y2^2*t^2-Y1^2*t^2*Y2*q*nu*II-Y1^2*t*Y2^2*q*II+2*Y1^2*t*Y2^2*q*nu*II-Y1*t^2*Y2^2*q*nu*II+Y1*t^2*Y2*q*nu-Y1*t^2*Y2*q-Y1*t...

>    nops(res);

7

>    seq(op(i,res),i=1..6);;

(-Y2+Y1)^4, Y1^4, t^6, q^2, nu^4, (nu-1)^2

>    e10:=map(factor,collect(op(1,op(7,res)),[Y1,Y2],distributed));

e10 := -t*(t*q*nu*II-2*nu-q+q*nu+2)*Y2^2*Y1-t*(t*q*nu*II-2*nu-q+q*nu+2)*Y1^2*Y2-t^2*(nu-1)*Y1^2-t^2*(nu-1)*Y2^2+(-q-nu-t^3*q^2*nu-t*q*II+q*nu+1+2*t*q*nu*II)*Y1^2*Y2^2+t^2*(-2+q)*(nu-1)*Y1*Y2
e10 := -t*(t*q*nu*II-2*nu-q+q*nu+2)*Y2^2*Y1-t*(t*q*nu*II-2*nu-q+q*nu+2)*Y1^2*Y2-t^2*(nu-1)*Y1^2-t^2*(nu-1)*Y2^2+(-q-nu-t^3*q^2*nu-t*q*II+q*nu+1+2*t*q*nu*II)*Y1^2*Y2^2+t^2*(-2+q)*(nu-1)*Y1*Y2

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));

(nu-1)*t^2*(y1^2+y2^2-(-2+q)*y1*y2)+t*y1*y2*(y1+y2)*(t*q*nu*II+(-2+q)*(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);

0

>    eq14:=collect(numer(subs(Y1=1/Z1,Y2=1/Z2,e10)),[Z1,Z2],factor);

eq14 := -t^2*(nu-1)*Z1^2+(t^2*(-2+q)*(nu-1)*Z2-t*(t*q*nu*II-2*nu-q+q*nu+2))*Z1-t^2*(nu-1)*Z2^2-t*(t*q*nu*II-2*nu-q+q*nu+2)*Z2-q-nu-t^3*q^2*nu-t*q*II+q*nu+1+2*t*q*nu*II
eq14 := -t^2*(nu-1)*Z1^2+(t^2*(-2+q)*(nu-1)*Z2-t*(t*q*nu*II-2*nu-q+q*nu+2))*Z1-t^2*(nu-1)*Z2^2-t*(t*q*nu*II-2*nu-q+q*nu+2)*Z2-q-nu-t^3*q^2*nu-t*q*II+q*nu+1+2*t*q*nu*II

>    collect(subs(Z1=V1+a,Z2=V2+a,eq14),[V1,V2],factor);

-t^2*(nu-1)*V1^2+(t^2*(-2+q)*(nu-1)*V2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2))*V1-t^2*(nu-1)*V2^2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2)*V2-4*t^2*a^2*nu+4*t^2*a^2-...
-t^2*(nu-1)*V1^2+(t^2*(-2+q)*(nu-1)*V2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2))*V1-t^2*(nu-1)*V2^2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2)*V2-4*t^2*a^2*nu+4*t^2*a^2-...
-t^2*(nu-1)*V1^2+(t^2*(-2+q)*(nu-1)*V2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2))*V1-t^2*(nu-1)*V2^2-t*(4*t*a*nu+t*a*q-t*a*q*nu-4*t*a+t*q*nu*II-2*nu-q+q*nu+2)*V2-4*t^2*a^2*nu+4*t^2*a^2-...

>    asol:=factor(solve(subs(V2=0,coeff(%,V1,1)),a));

asol := (t*q*nu*II-2*nu-q+q*nu+2)/(q-4)/(nu-1)/t

>    eq15:=collect(subs(Z1=V1+asol,Z2=V2+asol,-eq14/t^2/(nu-1)),[V1,V2],factor);

eq15 := V1^2+(2-q)*V2*V1+V2^2+(1-2*nu-8*t*nu*II+t^3*q^2*nu^2-t^3*q^2*nu-4*t^3*q*nu^2+4*t^3*q*nu-t*q*II+4*t*II+nu^2+t*q*nu*II+4*t*nu^2*II+t^2*q*nu^2*II^2)/t^2*q/(nu-1)^2/(q-4)

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);

CC := t^2*q*(1+beta)^2*II^2+beta*(4*beta+q)*t*II+q*beta*(1+beta)*(q-4)*t^3+beta^2

>