Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How to design algebraic function with particular ramified covering?

Loading...

Similar Threads - design algebraic function | Date |
---|---|

A Characterizing the adjoint representation | Dec 20, 2017 |

A Diagonalization of adjoint representation of a Lie Group | Dec 13, 2017 |

Help Re-Designing A Curve Using X Y Z Co-ordinates | Mar 12, 2012 |

Polygons can be designated as N-gons. What about vertices? | Jan 19, 2012 |

Affine plane, block design | Jan 3, 2011 |

**Physics Forums - The Fusion of Science and Community**