Generators in the field of hyperelliptic functions

TL;DR

This paper explores the structure of hyperelliptic function fields, establishing their generators are algebraically independent and comparing modern and classical definitions within the context of Kleinian functions.

Contribution

It demonstrates that the hyperelliptic function field for a family of curves of genus g is generated by 3g algebraically independent functions, clarifying their structure.

Findings

The hyperelliptic function field is isomorphic to a rational function field with 3g generators.

There are no algebraic relations among these generators.

The work compares modern and classical definitions of hyperelliptic functions.

Abstract

We consider the field of hyperelliptic functions defined for a family of hyperelliptic curves as rational functions in some special functions from Kleinian functions theory. We compare our definition with the classical one. We provide details and references for the result that the field of hyperelliptic functions for a family of hyperelliptic curves of genus $g$ is isomorphic to the field of rational functions with $3g$ generators. The main result of the present work is that there are no algebraic relations between these generators.