The case q=3 (m=6,k=1) : an equation with a single catalytic variable (Section 13.1)

>    T:=proc(m) subs(cos(x)=x,expand(cos(m*x))): end:

>    T(6);

32*x^6-48*x^4+18*x^2-1

>    m:=6: k:=1: 2+2*cos(2*k*Pi/m);

3

>    I2:=normal(subs(C=CC,II=I1,q=3,yinv=1/y,M(1,y)=M(y),normal(sqrt(C)^m*subs(x=xsubs,T(m))))):

>    indets(I2);

{nu, t, w, y, M(y)}

>    factor(series(normal(I2),y=1,1));

series(1/2*t^6*(nu^2+4*nu+1)*(2*nu^2+2*nu-1)*(nu^2-2*nu-2)*(y-1)^(-6)+O((y-1)^(-5)),y=-(-1),-5)

>    eqinv:=subs(M(1,y)=M(y),q=3,I2-add(c[i]*I1^i,i=0..6)):

>    indets(eqinv);

{nu, t, w, y, c[0], c[1], c[2], c[3], c[4], c[5], c[6], M(y)}

>   

>   

Determination of the series c_i. An equation with a single catalytic variable and 4 additional unknown series, M(1), M'(1),...,M'''(1)

>   

>    eqinvM;

162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...
162*nu*w^3*y^4*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(1)^4+(162*nu*w^3*y^5*(y-1)^3*(2*nu+1)*(nu+2)*(nu+1)*t^4*M(y)+54*nu*w^2*y^4*(7*y-6)*(y-1)^2*(2*nu+1)*(nu+2)*(nu+1)*t^4+54*w^2*y^3*(y-1)^3*(-28*nu^2*y-...

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The case nu=0 and w=1 (properly 3-coloured maps by edges) : the equation with one catalytic variable

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