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Hi, given the algebraic function:
[tex]f(z,w)=a_n(z)w^n+a_{n1}(z)w^{n1}+\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)=(w1)(w2)^2(w3)^3z=0[/tex]
That one has a single sheet manifold, a doublesheet and a triplesheet 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(wg(z)\right)\left(wh(z^{1/2})\right)\left(wu(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 singlevalued terms, i.e., gold and purple are multivalued. 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,
[tex]f(z,w)=a_n(z)w^n+a_{n1}(z)w^{n1}+\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)=(w1)(w2)^2(w3)^3z=0[/tex]
That one has a single sheet manifold, a doublesheet and a triplesheet 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(wg(z)\right)\left(wh(z^{1/2})\right)\left(wu(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 singlevalued terms, i.e., gold and purple are multivalued. 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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