Hello, I have a somewhat math and Maple question I'm hoping some can help with.
I have this curve,
and if I solve this system (numerically);
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;
and convert the 6 coordinates and curve, I'll get everything in terms of (u,v) coordinates.
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);
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,
I believe I am doing something wrong though bcause every odd power is zero.
Thank you a lot for any suggestions and/or help,
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...