The case q=2 et nu=0 (Section 2)

The functional equation (we only count edges, so that we set w=1)

>    eqM2:=subs(nu=0,q=2,w=1,eqM);;

eqM2 := 1+x*y*t*(y+1)*M(x,y)*M(1,y)-x*y*t*M(x,y)*M(x,1)-x*y*t*(x*M(x,y)-M(1,y))/(x-1)+x*y*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y)

>    Ker2:=-coeff(eqM2,M(x,y));

Ker2 := -x*y*t*(y+1)*M(1,y)+x*y*t*M(x,1)+x^2*y*t/(x-1)-x*y^2*t/(y-1)+1

>    R2:=coeff(eqM2,M(x,y),0);

R2 := 1+x*y*t*M(1,y)/(x-1)-x*y*t*M(x,1)/(y-1)

>    subs(x=1+s*t,Ker2*(y-1));

(-(1+t*s)*y*t*(y+1)*M(1,y)+(1+t*s)*y*t*M(1+t*s,1)+(1+t*s)^2*y/s-(1+t*s)*y^2*t/(y-1)+1)*(y-1)

>    factor(subs(t=0,%));

(s+y)*(y-1)/s

Expansion of Y_1 and Y_2

>    n:=6:
res:=expand(convert(subs(M(1,y)=subs(x=1,Mser(n)),M(x,1)=subs(y=1,Mser(n)), x=1+t*s,q=2,nu=0,numer(Ker2)),polynom)):

>    with(gfun):

>    algeqtoseries(res,t,y,3,true);

[series(1+s/(1+s)*t+s^2*(1+3*s+s^2)/(1+s)^3*t^2+O(t^3),t,3), series((-s)+s^2*(2+s^2+2*s)/(1+s)*t+s^2*(w-3*s-9*s^2-8*s^5-2*s^6-15*s^4-16*s^3+s^6*w+2*s^5*w-s^3*w+3*s*w+2*s^2*w)/(1+s)^3*t^2+O(t^3),t,3)]
[series(1+s/(1+s)*t+s^2*(1+3*s+s^2)/(1+s)^3*t^2+O(t^3),t,3), series((-s)+s^2*(2+s^2+2*s)/(1+s)*t+s^2*(w-3*s-9*s^2-8*s^5-2*s^6-15*s^4-16*s^3+s^6*w+2*s^5*w-s^3*w+3*s*w+2*s^2*w)/(1+s)^3*t^2+O(t^3),t,3)]

>   

>   

Construction of the invariants (performed for general q and nu further down in the session)

The two invariants

>    Idef; Jdef;

2*t*y*M(1,y)+(y-1)/y+t*y/(y-1)

(II^2-2*II+2*t+2)^2-8/y^2*(II^2-2*II+2*t+2)+8/y^4

The  invariant equation (12).

>    eqinv:=Jdef-add(C[r]*II^r,r=0..4);

eqinv := (II^2-2*II+2*t+2)^2-8/y^2*(II^2-2*II+2*t+2)+8/y^4-C[0]-C[1]*II-C[2]*II^2-C[3]*II^3-C[4]*II^4

Or, in terms of  M(1,y)

>    eqinvM:=subs(II=Idef,eqinv):

We now determine C_0, ..., C_4 by an expansion of the invariant equation around y=1.

>    factor(series(eqinvM,y=1,1));

series((-t^4*(-1+C[4]))*(y-1)^(-4)+O((y-1)^(-3)),y=-(-1),-3)

>    factor(series(subs(C[4]=1,eqinvM),y=1,2));

series((-t^3*(4+C[3]))*(y-1)^(-3)+(-t^2*(8*t+C[2]+3*t*C[3]+6*t*C[3]*M(1,1)+24*t*M(1,1)))*(y-1)^(-2)+O((y-1)^(-1)),y=-(-1),-1)

>    factor(series(subs(C[4]=1,C[3]=-4,eqinvM),y=1,2));

series(t^2*(4*t-C[2])*(y-1)^(-2)+O((y-1)^(-1)),y=-(-1),-1)

>    factor(series(subs(C[4]=1,C[3]=-4,C[2]=4*t,eqinvM),y=1,3));

series(t*(8-C[1]+8*t)*(y-1)^(-1)+O(1),y=-(-1),0)

>    factor(series(subs(C[4]=1,C[3]=-4,C[2]=4*t,C[1]=8+8*t,eqinvM),y=1,4));

series((-4-40*t-C[0]+32*t^2*M(1,1)-4*t^2)+O((y-1)),y=-(-1),1)

>    coeff(%,y-1,0);

-4-40*t-C[0]+32*t^2*M(1,1)-4*t^2

The equation on M(1,y)

>    collect(numer(factor(subs(C[4]=1,C[3]=-4,C[2]=4*t,C[1]=8+8*t,C[0]=-4-40*t-4*t^2+32*t^2*M(1,1),eqinvM)))/t/32,M,factor);

y^2*t*(y-1)*(y+1)*M(1,y)^2+(y^2*t+1-y^2)*M(1,y)-t*M(1,1)*y^2+(y-1)*(y+1)

Compare with (6):

>    eqref:=1+t*y^2*M(1,y)^2+t*y^2*(M(1,y)-M(1,1))/(y^2-1)-M(1,y);

eqref := 1+t*y^2*M(1,y)^2+t*y^2*(M(1,y)-M(1,1))/(-1+y^2)-M(1,y)

>    factor(%%/eqref);

(y-1)*(y+1)

>