Topology of real algebraic curves near the non-singular tropical limit

TL;DR

This paper advances the understanding of real algebraic curves near tropical limits, providing new criteria for their topology and constructing counterexamples to Ragsdale's conjecture.

Contribution

It introduces new patchworking criteria for non-singular real algebraic curves in toric surfaces based on tropical geometry.

Findings

New criteria for patchworking real algebraic curves with prescribed components

Conditions for curves with only ovals in their real part

Counterexamples to Ragsdale's conjecture

Abstract

In the 1990's, Itenberg and Haas studied the relations between combinatorial data in Viro's patchworking and the topology of the resulting non-singular real algebraic curves in the projective plane. Using recent results from Renaudineau and Shaw on real algebraic curves near the non-singular tropical limit, we continue the study of Itenberg and Haas inside any non-singular projective toric surface. We give new Haas' like criteria for patchworking non-singular real algebraic curves with prescribed number of connected components, in terms of twisted edges on a non-singular tropical curve. We then obtain several sufficient conditions for patchworking non-singular real algebraic curves with only ovals in their real part. One of these sufficient conditions does not depend on the ambient toric surface. In that case, we count the number of even and odd ovals of those curves in terms of the dual subdivision, and construct some new counter-examples to Ragsdale's conjecture.