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How to design algebraic function with particular ramified covering? 
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#1
Nov1312, 01:08 PM

P: 1,666

Hi,
I wish to study (nontrivial) 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 z4 z^26 z^3)w+(10 z)w^2+(3 z^26 z^3)w^3+(9 z^2+4 z^4)w^4+(65 z+z^3)w^5[/tex] has a 1cycle and 4cycle 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 2cycle 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 1cycle and a 4cycle 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(z1,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 nontrivial algebraic function of degree at least 5 in [itex]p_n(z)[/itex] and w so that it has a nontrivial ramified covering above several of it's nonzero singular points and by nontrivial, I mean coverings which are more ramified than a set of singlecover branches with one 2cycle branch? Thanks, Jack 


#2
Nov2112, 09:17 AM

P: 1,666

Hey guys. Turns out to be a trival matter. Let's take a simple example:
[tex]f(z,w)=p_0+p_1 w+p_2 w^2+p_3 w^3[/tex] and suppose we wish to have it fullyramify at the points [itex]z=1,1[/itex]. A sufficient condition for full ramification of the translated function [itex]f(z\pm 1,w)[/itex], is for the lower Newtonleg of it's first polygon to have a slope of [itex]1/3[/itex]. Thus, one possibility is: [tex]f(z\pm1,w)=(z+q_1)+(z+q_2)w+(z+q_3)+(a+q_4)w^3[/tex] where each [itex]q_i[/itex] is a suitable polynomial in [itex]z[/itex]. Take [itex]p_0(z)=a+bz+cz^2[/itex]. Then one solution is: [itex]a+b(z\pm 1)+c(z\pm 1)^2=\alpha z+\beta z^2[/itex] or [itex]a\pm b+c=0[/itex]. Let's take b=0, a=1 and c=1 so that for the first term, [itex]p_0(z)=1z^2[/itex]. We can do the same for the others and arrive at a suitable function: [tex]f(z,w)=(1z^2)+(1z^2)w+(1z^2)w^2+z w^3[/tex] with the desired ramification geometry at the points z=1 and 1. 


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