Inflection Points of Real and Tropical Plane Curves

TL;DR

This paper demonstrates that Viro's patchworking can produce real algebraic curves with the maximum number of inflection points, using tropical geometry techniques to analyze their properties.

Contribution

It introduces a tropical approach to studying inflection points of real algebraic curves, establishing maximal inflection points via tropical modifications.

Findings

Maximally inflected real algebraic M-curves realize many isotopy types.

Tropical limits of inflection points can be effectively studied using tropical modifications.

Viro's patchworking can produce curves with the maximal number of real inflection points.

Abstract

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.