Register to reply

How to design algebraic function with particular ramified covering?

by jackmell
Tags: algebraic, covering, design, function, ramified
Share this thread:
Nov13-12, 01:08 PM
P: 1,666

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:


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?

Phys.Org News Partner Science news on
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
Nov21-12, 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 fully-ramify 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 Newton-leg of it's first polygon to have a slope of [itex]1/3[/itex]. Thus, one possibility is:


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)=1-z^2[/itex]. We can do the same for the others and arrive at a suitable function:

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

with the desired ramification geometry at the points z=-1 and 1.

Register to reply

Related Discussions
Reducing algebraic function to DE Differential Equations 1
Factorizing an Algebraic Function Precalculus Mathematics Homework 2
Algebraic Function Question Precalculus Mathematics Homework 4
Algebraic topology, groups and covering short, exact sequences Differential Geometry 1
Algebraic function help Precalculus Mathematics Homework 1