Separating semigroup of hyperelliptic curves and of genus 3 curves

TL;DR

This paper characterizes the separating semigroup of hyperelliptic and genus 3 real algebraic curves, providing a detailed description of the degrees of separating functions on their connected components.

Contribution

It offers the first explicit description of the separating semigroup for hyperelliptic and genus 3 curves, advancing understanding of real algebraic curve coverings.

Findings

Explicit description of the separating semigroup for hyperelliptic curves

Explicit description of the separating semigroup for genus 3 curves

Enhanced understanding of real algebraic curve coverings

Abstract

A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent paper M. Kummer and K. Shaw defined the separating semigroup of $C$ as the set of all sequences $(d_1(f),\dots,d_n(f))$ where $f$ is a separating function and $d_i(f)$ is the degree of the restriction of $f$ to $A_i$. We describe the separating semigroup for hyperelliptic curves and for genus 3 curves.