Real inflection points of real hyperelliptic curves

TL;DR

This paper investigates the real inflection points of hyperelliptic curves using Viro's patchworking and Berkovich spaces, offering a simpler alternative to existing limit linear series methods for curves in toric surfaces.

Contribution

It introduces a new approach combining Viro's patchworking and Berkovich spaces to analyze real inflection points, simplifying previous complex methods.

Findings

Explicit description of real inflection points on hyperelliptic curves.

A new method that simplifies analysis compared to limit linear series.

Application of Berkovich spaces in real algebraic geometry.

Abstract

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.