Tropical Ideals

TL;DR

This paper introduces tropical ideals in the semiring of tropical polynomials, establishing their algebraic properties and their role in defining tropical varieties as finite polyhedral complexes, thus advancing tropical geometry foundations.

Contribution

It defines tropical ideals, extends classical ideal concepts to tropical geometry, and demonstrates their structural properties and applications to tropical varieties.

Findings

Tropical ideals include tropicalizations of classical ideals.

Every tropical ideal has an associated finite polyhedral complex variety.

Tropical ideals satisfy the ascending chain condition and weak Nullstellensatz.

Abstract

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing [Giansiracusa-Giansiracusa]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.