Tropical curves with parallel rays

TL;DR

This paper introduces a new category of abstract tropical curves with parallel rays, addressing a flaw in traditional definitions, and establishes categorical equivalences with semifields over the tropical semifield.

Contribution

It defines and characterizes tropical curves with parallel rays and proves categorical equivalences with semifields, extending the algebraic understanding of tropical geometry.

Findings

Categorical equivalence between tropical curves with parallel rays and semifields over the tropical semifield.

New notion of abstract tropical curves with parallel rays introduced and characterized.

Geometric notions like weights and balancing condition translated into algebraic terms.

Abstract

In the previous works, the rational function semifields of abstract tropical curves were characterized. In this paper, we give a contravariant categorical equivalence between the category of abstract tropical curves with morphisms and the category of semifields over the tropical semifield $\boldsymbol{T}$ characterized above with $\boldsymbol{T}$-algebra homomorphisms. The characterization tells us that the traditional definition of abstract tropical curves has a fatal flaw such that we are never able to deal with parallel rays, unlike the traditional tropical curves, which generally admit them. To address this flaw, we introduce a new notion of abstract tropical curves with parallel rays. Then we define the rational function semifields of these curves and give a characterization of them, and a variant of the categorical equivalence between their categories with a suitable notion of morphisms between these curves. Under the categorical equivalences, we translate several geometric notions for traditional or abstract tropical curves (with parallel rays) into algebraic ones, including weights on edges and the balancing condition.