Congruences and coordinate semirings of tropical varieties

TL;DR

This paper introduces two intrinsic algebraic definitions of tropical varieties, connecting them with classical algebraic geometry concepts and exploring their relation to the dimension of affine tropical varieties.

Contribution

It presents novel algebraic frameworks for tropical varieties using coordinate semirings and layered structures, inspired by the classical Zariski correspondence.

Findings

Two algebraic definitions of tropical varieties are proposed.

The connection between these definitions and the dimension of affine tropical varieties is established.

The work bridges tropical geometry with classical algebraic concepts.

Abstract

In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, and the second based on the layered structure. We tie them to tropical geometry, especially in connection with the dimension of an affine variety.