Separating semigroup of hyperelliptic curves and of genus 3 curves

Abstract:

A rational function on a real algebraic curve $C$ is said to be separating if it takes real values only at real points. Such a function gives rise to a covering $\mathbb {R} C\to \mathbb {R}\mathbb {P}^1$. Let $A_1,\dots ,A_n$ be connected components of $\mathbb R C$. 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 $d_i$ is the degree of the restriction of $f$ to $A_i$.

Here, the separating semigroups for hyperelliptic curves and for genus 3 curves are described.