Hi, given the algebraic function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f(z,w)=a_n(z)w^n+a_{n-1}(z)w^{n-1}+\cdots+a_0(z)=0[/tex]

how can I determine the geometry of it's underlying Riemann surfaces? For example, here's a contrived example:

[tex]f(z,w)=(w-1)(w-2)^2(w-3)^3-z=0[/tex]

That one has a single sheet manifold, a double-sheet and a triple-sheet manifold as best as I can describe it and illustrated by the real part of w(z) shown below and will of course give rise to the expansion

[tex]f(z,w)=\left(w-g(z)\right)\left(w-h(z^{1/2})\right)\left(w-u(z^{1/3})\right)=0[/tex]

or:

[tex]f(z,w)=\left(w-\text{red}\right)\left(w-\text{gold}\right)\left(w-\text{purple}\right)=0[/tex]

which is my interest in this problem and I've not split up the factoring into separate single-valued terms, i.e., gold and purple are multi-valued. However, is there a method to determine this manifold geometry for the general case just by an analysis of the function [itex]f(z,w)[/itex]? I know the Newton Polygon procedure can determine this indirectly but I believe that has limitations and will not work for certain functions. I was wondering is there is another way. That is, given [itex]f(z,w)[/itex], is there some (exact) algebraic operation on it I can perform and obtain the sheet orders which in the contrived case above would be 1,2, and 3?

Thanks,

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# Determining Riemann surface geometry of algebraic functions

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