# Question:Problem with power series

## Question:Problem with power series

Maple 2015

Hello, I have a somewhat math and Maple question I'm hoping some can help with.

I have this curve,

curve:=alpha^2*beta^10-alpha^2*beta^9+4*alpha*beta^7-2*alpha*beta^6-2*alpha*beta^5-alpha*beta^4+alpha*beta^3-2*beta+1;

and if I solve this system (numerically);

sol:= solve([curve=0,diff(curve,beta)=0],[alpha,beta]);

I get 6 special points (8 actually but two are critical). So I'll refer to them by subscript "i".

If I then do a coordinate transformation by;

alpha=exp(u)

beta=exp(v)

and convert the 6 coordinates and curve, I'll get everything in terms of (u,v) coordinates.

curve2:=subs([alpha=exp(u),beta=exp(v)],curve);

soluv:=map(ln,sol);  (this is just pseudo - I don't know how to do it this way)

So now the 6 points are referred to by (u_i,v_i).

Next, I want to expand this curve locally around these six points, using the following (where "z" is the local coordinate);

u -> u_i + z^2

v -> v_i + sum(a_j*z^j,j=1..n)

where n is reasonable, though around 15.

curve3(i):=subs(u=z^2 + cat(Ubp,i),curve2);

curve4(i,n):=subs(v=cat(Vbp,i)+sum(a[i,j]*z^j,j=1..n),curve3(i));

Here, I'm not sure of the pros/cons of cat() vs a[i,j]....

Anyway, I'll then have an equation in terms of only the local coordinate "z". If I then solve each coefficient of "z", at each order, I should then be able to determine the power series v(z). I reason that since the curve is initially equal to zero, that every non-zero power of z will have a coefficient/equation (in terms of unknowns a[i,j]) that should be equal to zero.

This is analagous to solving differential equations with power series...

However, I'm a little lost in implementing this,

I am currently trying, for instance,

l1:= series(curve4(1,10),z,0,9);

e1:= seq(coeff(l1,z,i),i=1..4):

s1:=seq(a[1,j],j=1..4):

sol:=solve(e1,s1);

I believe I am doing something wrong though bcause every odd power is zero.

Thank you a lot for any suggestions and/or help,

sbh

PS>

Ubp and Vbp are those 6 points - I just kept them as symbols initially because I was still getting odd-powered coefficients as zero, regardless of their actual values. Hence a little bit of the "math" side of the problem... ﻿