Hyperbolic plane curves near the non-singular tropical limit

TL;DR

This paper characterizes when real algebraic curves near a non-singular tropical limit are hyperbolic, introducing new tools in real tropical intersection theory and analyzing hyperbolicity loci, especially for honeycombs.

Contribution

It provides necessary and sufficient conditions for hyperbolicity near the tropical limit and develops new real tropical intersection tools.

Findings

Characterization of hyperbolic curves near the tropical limit

Introduction of tropical hyperbolicity and signed hyperbolicity loci

Description of hyperbolicity loci for honeycombs in terms of twisted edges

Abstract

We determine necessary and sufficient conditions for real algebraic curves near the non-singular tropical limit to be hyperbolic with respect to a point, thus generalising Speyer's classification of stable curves near the tropical limit. In order to obtain the conditions, we develop tools of real tropical intersection theory. We introduce the tropical hyperbolicity locus and the signed tropical hyperbolicity locus of a real algebraic curve near the non-singular tropical limit. In the case of honeycombs, we characterise the tropical hyperbolicity locus in terms of the set of twisted edges on the tropical limit.