Geometry of tropical extensions of hyperfields

TL;DR

This paper explores the geometric properties of tropical extensions of hyperfields, introducing enriched valuations and establishing hyperfield analogues of key theorems in tropical geometry, leading to new insights into tropical varieties.

Contribution

It develops a framework of enriched valuations for hyperfield homomorphisms and proves hyperfield versions of Kapranov's theorem and the Fundamental theorem of tropical geometry.

Findings

Hyperfield analogues of Kapranov's theorem established

Introduction of fine tropical varieties and their structure theorem

Identification of algebraically closed hyperfield homomorphisms

Abstract

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov's theorem and the Fundamental theorem of tropical geometry. Utilising these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.