Patchworking the Log-critical locus of planar curves

TL;DR

This paper develops a patchworking theorem for the Log-critical locus of algebraic curves in the complex torus, enabling the construction of curves with prescribed Log-critical properties and analyzing Log-inflection points in families.

Contribution

It introduces a Viro-style patchworking theorem for the Log-critical locus and generalizes existing results on Log-inflection points in tropical geometry.

Findings

Established a patchworking theorem for Log-critical loci.

Proved the existence of curves with smooth connected Log-critical locus.

Generalized a theorem on the tropical limit of Log-inflection points.

Abstract

We establish a patchworking theorem \`a la Viro for the Log-critical locus of algebraic curves in $(\mathbb{C}^*)^2$. As an application, we prove the existence of projective curves of arbitrary degree with smooth connected Log-critical locus. To prove our patchworking theorem, we study the behaviour of Log-inflection points along families of curves defined by Viro polynomials. In particular, we prove a generalisation of a theorem of Mikhalkin and the second author on the tropical limit of Log-inflection points.