Back to Tutte's original equation (Section 14.2)

We start from the equation for Q that involves t, z and w (the first equation of the session)

>    eqQ;

-Q(x,y)+1+z*t/y*(Q(x,y)-1-y*Q1(x))+q*y^2*w*t*Q(x,y)*Q(0,y)+x*y*z*t*Q1(x)*Q(x,y)+x*z*t*(Q(x,y)-1)+(nu-1)*(y*z*t*Q(x,y)*(2*x*Q1(x)+Q2(x))+y^2*w*t/(1-x*nu*z*t)*Q(0,y)*Q(x,y)+w*t*y/(1-x*nu*z*t)/x*(Q(x,y)-Q...
-Q(x,y)+1+z*t/y*(Q(x,y)-1-y*Q1(x))+q*y^2*w*t*Q(x,y)*Q(0,y)+x*y*z*t*Q1(x)*Q(x,y)+x*z*t*(Q(x,y)-1)+(nu-1)*(y*z*t*Q(x,y)*(2*x*Q1(x)+Q2(x))+y^2*w*t/(1-x*nu*z*t)*Q(0,y)*Q(x,y)+w*t*y/(1-x*nu*z*t)/x*(Q(x,y)-Q...

The equation of Lemma 31

>    unassign('R'):eqQR:=1+Q(x,y)/Q(0,y)*R(x,Y)-Q(x,y);

eqQR := 1+Q(x,y)/Q(0,y)*R(x,Y)-Q(x,y)

>    Ydef:=y*Q(0,y);

Ydef := y*Q(0,y)

We now constrcut an equation for R

>    subs(x=0,eqQR);

1+R(0,Y)-Q(0,y)

>    Q0ysol:=solve(%,Q(0,y));

Q0ysol := 1+R(0,Y)

>    Qxysol:=solve(subs(Q(0,y)=Q0ysol,eqQR),Q(x,y));

Qxysol := (1+R(0,Y))/(1-R(x,Y)+R(0,Y))

>    subs(Q(0,0)=1, Q(x,0)=1,R(x,0)=0,series(subs(Y=Ydef,eqQR),y,1));

series(O(y),y,1)

>    subs(Q(0,0)=1, Q(x,0)=1,R(x,0)=0,series(subs(Y=Ydef,eqQR),y,2));

series((D[2](R)(x,0)-D[2](Q)(x,0))*y+O(y^2),y,2)

>    normal(subs(Q(0,0)=1, Q(x,0)=1,R(x,0)=0,D[2](R)(x,0)=D[2](Q)(x,0),series(subs(Y=Ydef,eqQR),y,3)));

series((1/2*D[2,2](R)(x,0)+D[2](Q)(x,0)^2-1/2*D[2,2](Q)(x,0))*y^2+O(y^3),y,3)

>    eqR0:=subs(Q(x,y)=Qxysol, Q(0,y)=Q0ysol, y=Y/Q0ysol, Q1(x)=R1(x), Q2(x)=R2(x)+R1(x)^2,eqQ);

eqR0 := -(1+R(0,Y))/(1-R(x,Y)+R(0,Y))+1+z*t/Y*(1+R(0,Y))*((1+R(0,Y))/(1-R(x,Y)+R(0,Y))-1-Y/(1+R(0,Y))*R1(x))+q*Y^2*w*t/(1-R(x,Y)+R(0,Y))+x*Y*z*t*R1(x)/(1-R(x,Y)+R(0,Y))+x*z*t*((1+R(0,Y))/(1-R(x,Y)+R(0,...
eqR0 := -(1+R(0,Y))/(1-R(x,Y)+R(0,Y))+1+z*t/Y*(1+R(0,Y))*((1+R(0,Y))/(1-R(x,Y)+R(0,Y))-1-Y/(1+R(0,Y))*R1(x))+q*Y^2*w*t/(1-R(x,Y)+R(0,Y))+x*Y*z*t*R1(x)/(1-R(x,Y)+R(0,Y))+x*z*t*((1+R(0,Y))/(1-R(x,Y)+R(0,...
eqR0 := -(1+R(0,Y))/(1-R(x,Y)+R(0,Y))+1+z*t/Y*(1+R(0,Y))*((1+R(0,Y))/(1-R(x,Y)+R(0,Y))-1-Y/(1+R(0,Y))*R1(x))+q*Y^2*w*t/(1-R(x,Y)+R(0,Y))+x*Y*z*t*R1(x)/(1-R(x,Y)+R(0,Y))+x*z*t*((1+R(0,Y))/(1-R(x,Y)+R(0,...

>    indets(%);

{Y, nu, q, t, w, x, z, R(0,Y), R(x,Y), R1(x), R2(x)}

>    eqR1:=-op(2,factor(numer(subs(Y=y,eqR0))));

eqR1 := y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*n...
eqR1 := y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*n...
eqR1 := y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*n...
eqR1 := y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*n...
eqR1 := y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*n...

>    subs(t=0,eqR1);

-y*x*R(x,y)

>    eqR2:=eqR1/x/y;

eqR2 := (y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*...
eqR2 := (y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*...
eqR2 := (y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*...
eqR2 := (y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*...
eqR2 := (y*x^2*R(x,y)*nu*z*t-x^2*z^2*t^2*R(x,y)*R(0,y)*nu+x^2*z^2*t^2*y*R1(x)*nu+x*z*t*y*R1(x)*R(x,y)-x*z*t*y*R1(x)*R(0,y)+x^3*y^2*z^2*t^2*R1(x)*nu-x^3*z^2*t^2*R(x,y)*y*nu+2*y^2*t*nu*x^2*z*R1(x)+y^2*t*...

Version of the paper

>    papier:=-R(x,Y)+ y^2*w*t*(1-x*nu*z*t)*q+(nu-1)*w*t*y^2
 +x*z*t*(1+nu-x*nu*z*t)*R(x,Y)+x*z*t*Y*(1-x*nu*z*t)*R1(x)+
z*t*(1-x*nu*z*t)*R1(x)*R(x,Y)  +z*t*(1-x*nu*z*t)*(1+R(0,Y))* (R(x,Y)-Y*R1(x))/Y
+(nu-1)* Y*z*t*(1-x*nu*t*z)*(R2(x)+R1(x)^2+2*x*R1(x))
+(nu-1)*w*t*Y*(R(x,Y)-R(0,Y))
/x;

papier := -R(x,Y)+y^2*w*t*(1-x*nu*z*t)*q+(nu-1)*w*t*y^2+x*z*t*(1+nu-x*nu*z*t)*R(x,Y)+x*z*t*Y*(1-x*nu*z*t)*R1(x)+z*t*(1-x*nu*z*t)*R1(x)*R(x,Y)+z*t*(1-x*nu*z*t)*(1+R(0,Y))*(R(x,Y)-Y*R1(x))/Y+(nu-1)*Y*z*t...
papier := -R(x,Y)+y^2*w*t*(1-x*nu*z*t)*q+(nu-1)*w*t*y^2+x*z*t*(1+nu-x*nu*z*t)*R(x,Y)+x*z*t*Y*(1-x*nu*z*t)*R1(x)+z*t*(1-x*nu*z*t)*R1(x)*R(x,Y)+z*t*(1-x*nu*z*t)*(1+R(0,Y))*(R(x,Y)-Y*R1(x))/Y+(nu-1)*Y*z*t...

>    collect(factor(eqR2/subs(Y=y,papier)),R1(x),factor);

1

Non-separable maps of Q: set nu=0 in the previous equation. Then R1=0=S1

>    eqS:=factor(numer(subs(R=S,R1(x)=0,R2=S2,nu=0,eqR2)));

eqS := q*y^3*w*t*x-y^3*t*w*x-y^2*t*x*z*S2(x)-y*x*S(x,y)+x*z*t*S(x,y)-y^2*t*w*S(x,y)+y^2*t*w*S(0,y)+x^2*z*t*S(x,y)*y+x*z*t*S(x,y)*S(0,y)

We finally reach Tutte's series T. Denote X=1/(1-xz) (see lemma 33)

>    eqT:=numer(factor(subs(w=1,t=1,  S(x,y)=T(X,y)/q, S2(x)=T2(X)/q, S(0,y)=T(1,y)/q,x=(1-1/X)/z,X=x,eqS)));

eqT := q^3*y^3*x^2-q^3*y^3*x-y^3*x^2*q^2+y^3*x*q^2-y*T(x,y)*x*q+T(x,y)*x^2*z*q-T(x,y)*x*z*q-y^2*T(x,y)*x^2*z*q+y^2*T(1,y)*x^2*z*q+T(x,y)*y*q+T(x,y)*T(1,y)*x^2*z-T(x,y)*T(1,y)*x*z-y^2*T2(x)*x^2*z*q+y^2*...
eqT := q^3*y^3*x^2-q^3*y^3*x-y^3*x^2*q^2+y^3*x*q^2-y*T(x,y)*x*q+T(x,y)*x^2*z*q-T(x,y)*x*z*q-y^2*T(x,y)*x^2*z*q+y^2*T(1,y)*x^2*z*q+T(x,y)*y*q+T(x,y)*T(1,y)*x^2*z-T(x,y)*T(1,y)*x*z-y^2*T2(x)*x^2*z*q+y^2*...

Version de Tutte

>    eqTutte:=x*y^2*q*(q-1)+x*z/y/q*T(1,y)*T(x,y)+x*z*(T(x,y)-y^2*T2(x))/y-x^2*y*z*(T(x,y)-T(1,y))/(x-1)-T(x,y);

eqTutte := x*y^2*q*(q-1)+x*z/y/q*T(1,y)*T(x,y)+x*z*(T(x,y)-y^2*T2(x))/y-x^2*y*z*(T(x,y)-T(1,y))/(x-1)-T(x,y)

>    factor(eqT/eqTutte);

q*y*(x-1)

>