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;

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));

ser := series(1+2*t+9*t^2+54*t^3+378*t^4+2916*t^5+24057*t^6+208494*t^7+1876446*t^8+17399772*t^9+165297834*t^10+1602117468*t^11+15792300756*t^12+157923007560*t^13+1598970451545*t^14+16365932856990*t^15+...
ser := series(1+2*t+9*t^2+54*t^3+378*t^4+2916*t^5+24057*t^6+208494*t^7+1876446*t^8+17399772*t^9+165297834*t^10+1602117468*t^11+15792300756*t^12+157923007560*t^13+1598970451545*t^14+16365932856990*t^15+...

>    with(gfun):

>    seriestoalgeq(ser,G(t));

[-1/27+16/27*t+(1/27-2/3*t)*G(t)+t^2*G(t)^2, ogf]

Conjectural equation satisfied by G(1)

>    algG:=op(1,%);

algG := -1/27+16/27*t+(1/27-2/3*t)*G(t)+t^2*G(t)^2

>    with(algcurves):

>    factor(parametrization(subs(G(t)=G,algG),t,G,T));

[-(19*T-1)*(-1+4*T)/(53*T-2)^2, -(53*T-2)*(23*T-2)/(-1+4*T)^2]

After a few simplifications...

>    algS:=S-t*(1+3*S)^2;

algS := S-t*(1+3*S)^2

>    tS:=solve(algS,t);

tS := S/(1+3*S)^2

>    factor(subs(G(t)=G,t=tS,algG));factor([solve(%,G)]);

1/27*(3*S^2-2*S+G-1)*(27*G*S^2+1-6*S-27*S^2)/(1+3*S)^4

[-(1+3*S)*(S-1), 1/27*(1+3*S)*(9*S-1)/S^2]

Conjectural value of G(1) in terms of the series S

>    F0sol:=(1-S)*(1+3*S);

F0sol := (1-S)*(1+3*S)

>    eqFS:=factor(numer(subs(G=F,G1=F0sol,t=tS,eqG)));

eqFS := y-1+5*S*y-6*S+7*S^2*y-9*S^2+S*y^3*F^2-S*y^2*F^2+S*y^2*F+3*S^3*y-y*F+F-6*F*S*y+6*F*S-9*F*S^2*y+9*F*S^2

>    genus(eqFS,y,F);

0

Rational paramatrization of this curve

>    eqW:=W*(1+3*S)-y*(1+S*W+S*(S+1)*W^2);

eqW := W*(1+3*S)-y*(1+S*W+S*(S+1)*W^2)

>    yW:=solve(eqW,y);

yW := W*(1+3*S)/(1+S*W+S^2*W^2+S*W^2)

>    factor(subs(y=yW,eqFS));

-(1+3*S)^2*(S^2*W^3*F+W^3*F*S-S*W^2*F+1-W+S*W+S^2*W^2-S*W^3-S^2*W^3)*(1-F-2*S^3*W^3-S^2*W^3-S^2*W-S^3*W^2-S^4*W^3+W*F*S+S*W^2)/(1+S*W+S^2*W^2+S*W^2)^3

>    factor([solve(%,F)]);

[(S^2*W+S*W-1)*(1+S*W+S^2*W^2+S*W^2)/(S*W-1), (-1+W)*(1+S*W+S^2*W^2+S*W^2)/(S*W+W-1)/W^2/S]

The first solution is indeed a series in S (or t) with polynomial coefficients in y

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