Did some experiments including a large degree example and it is really tricky to get the matrix which computes the coefficients of the Belyi map using power series expansions to have good stability properties. You can fiddle with a lot of things (replace w <- c*w in the series, rescale x and y, etc.), and this can significantly change things like the condition number.
But I'm not sure we need any of this! The series are designed to behave when they are evaluated, and indeed when we work in precision 10^(-100) we'll get values of x and y that are pretty close to that amount of precision. (Even though the precision in the coefficients decays polynomially.)
In genus zero, no series are solved for: the Newton equations are set up using the ramification values in the usual way: we write A(x) + B(x) = C(x) with A,B,C factored, and we plug in the approximate roots.
I think we can do the same in the elliptic and hyperelliptic case. Because of the issue of the common zero, we may need just a tiny bit from the series, imposing a few additional linear equations, but that should be it. That should mean that we can get away with much less power series precision, which means the whole thing will run a lot lot faster.
Did some experiments including a large degree example and it is really tricky to get the matrix which computes the coefficients of the Belyi map using power series expansions to have good stability properties. You can fiddle with a lot of things (replace w <- c*w in the series, rescale x and y, etc.), and this can significantly change things like the condition number.
But I'm not sure we need any of this! The series are designed to behave when they are evaluated, and indeed when we work in precision 10^(-100) we'll get values of x and y that are pretty close to that amount of precision. (Even though the precision in the coefficients decays polynomially.)
In genus zero, no series are solved for: the Newton equations are set up using the ramification values in the usual way: we write A(x) + B(x) = C(x) with A,B,C factored, and we plug in the approximate roots.
I think we can do the same in the elliptic and hyperelliptic case. Because of the issue of the common zero, we may need just a tiny bit from the series, imposing a few additional linear equations, but that should be it. That should mean that we can get away with much less power series precision, which means the whole thing will run a lot lot faster.