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michaelmusty / Belyi

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Genus zero equations #22

Open jvoight opened 2 years ago

jvoight commented 2 years ago

Right now, in genus zero we only use the equations implied by A(x) - B(x) = C(x) where phi(x) = A(x)/B(x) and phi(x)-1 = C(x)/B(x).

The resulting ideal is far from reduced (see SV, https://math.dartmouth.edu/~jvoight/articles/compute-belyi-090519.pdf, Example 2.11), and a way to improve things is to apply the Atkin-Swinnerton-Dyer trick.

We should be able to get away with a lot less precision as input into Newton this way.

jvoight commented 2 years ago

Does it make any difference to factor the polynomials completely and insist that the roots are distinct? Of course we’ll multiply back together to get K-rationality of the coefficients, but Newton won’t care about that in the middle.