On rational functions whose normalization has genus zero or one

TL;DR

This paper classifies rational functions based on the genus of their Galois closure, providing a complete list for genus zero and a geometric description for genus one, advancing understanding of their algebraic and geometric properties.

Contribution

It offers a complete classification of rational functions with genus zero Galois closure and describes those with genus one, filling gaps in the understanding of their algebraic structure.

Findings

Complete list of rational functions with genus zero Galois closure

Geometric description of functions with genus one

Advances classification of rational functions by algebraic genus

Abstract

We give a complete list of rational functions $A$ such that the genus $g$ of the Galois closure of $\mathbb C(z)/\mathbb C(A)$ equals zero. We also provide a geometric description of $A$ for which $g=1.$