Generalising Kapranov's Theorem For Tropical Geometry Over Hyperfields

TL;DR

This paper extends Kapranov's theorem in tropical geometry by replacing the valuation map with hyperfield homomorphisms, broadening the theorem's applicability to more general algebraic structures.

Contribution

It generalizes Kapranov's theorem using hyperfield homomorphisms, including the complex tropical hyperfield, and outlines conditions for these homomorphisms to satisfy algebraic closure.

Findings

Generalization of Kapranov's theorem to hyperfield homomorphisms.

Analysis of the complex tropical hyperfield to the tropical hyperfield.

Conditions for hyperfield homomorphisms to satisfy algebraic closure.

Abstract

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation map, from a field to the real numbers union negative infinity, with a more general class of hyperfield homomorphisms, whose target is the tropical hyperfield and satisfy a relative algebraic closure condition. To provide an example of such a hyperfield homomorphism, the map from the complex tropical hyperfield to the tropical hyperfield is investigated. There is a brief outline of sufficient conditions for a hyperfield homomorphism to satisfy the relative algebraic closure condition.