Interlude : planar maps by guessing and checking (Section 13.1, continued)
> | eqG:=1+t*y^2*G^2+t*y*(y*G-G1)/(y-1)-G; |
Coefficients of G(y)
> | Gser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(G=Gser(n-1),G1=subs(y=1,Gser(n-1)),eqG+G),t,n+1))fi: end: |
Let us guess the coefficients (and value) of G(1)
> | ser:=subs(y=1,Gser(15)); |
> | with(gfun): |
> | seriestoalgeq(ser,G(t)); |
Conjectural equation satisfied by G(1)
> | algG:=op(1,%); |
> | with(algcurves): |
> | factor(parametrization(subs(G(t)=G,algG),t,G,T)); |
After a few simplifications...
> | algS:=S-t*(1+3*S)^2; |
> | tS:=solve(algS,t); |
> | factor(subs(G(t)=G,t=tS,algG));factor([solve(%,G)]); |
Conjectural value of G(1) in terms of the series S
> | F0sol:=(1-S)*(1+3*S); |
> | eqFS:=factor(numer(subs(G=F,G1=F0sol,t=tS,eqG))); |
> | genus(eqFS,y,F); |
Rational paramatrization of this curve
> | eqW:=W*(1+3*S)-y*(1+S*W+S*(S+1)*W^2); |
> | yW:=solve(eqW,y); |
> | factor(subs(y=yW,eqFS)); |
> | factor([solve(%,F)]); |
The first solution is indeed a series in S (or t) with polynomial coefficients in y
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