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I wish to study (non-trivial) algebraic functions which have varied cycles at a few places other than the origin. It's not hard to search for a particular covering at the origin. For example, the function

[tex]f(z,w)=(8 z^2)\text{}+(-10 z-4 z^2-6 z^3)w+(-10 z)w^2+(-3 z^2-6 z^3)w^3+(-9 z^2+4 z^4)w^4+(6-5 z+z^3)w^5[/tex]

has a 1-cycle and 4-cycle at the origin. I just randomly searched for one. However, it seems it's not so easy to find one with higher cycles than a 2-cycle outside the origin or maybe I'm missing something. I'd like to find some with more than one such point. I can certainly make a translation say:

[tex]g(z,w)=f(z+1,w)[/tex]

and now the function [itex]g(z,w)[/itex] will have a 1-cycle and a 4-cycle at the point z=1. But that's as far as I can go.

How would I design one with say two such cycle types say at the points -1 and 1?

I don't think just making the substitution g(z-1,w) would do it.

So is there a way to design an algebraic function [itex]f(z,w)[/itex] so that it will have a particular ramified covering at several points outside the origin or is there an easy way to search for ones other than just randomly?

Edit: Just thought of a more cioncise way of asking my question:

How can I find a non-trivial algebraic function of degree at least 5 in [itex]p_n(z)[/itex] and w so that it has a non-trivial ramified covering above several of it's non-zero singular points and by non-trivial, I mean coverings which are more ramified than a set of single-cover branches with one 2-cycle branch?

Thanks,

Jack

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# How to design algebraic function with particular ramified covering?

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