Varieties of prime tropical ideals and the dimension of the coordinate semiring
Dániel Joó, Kalina Mincheva
TL;DR
This work develops a rigorous bridge between tropical ideals and prime congruences, showing that every non-zero closed prime in ${\mathbb T}[x_1^{\pm1},\dots,x_n^{\pm1}]$ arises from a prime congruence via Bend, and that such ideals correspond to geometric primes when tropicalized. It proves that the tropical variety of a non-zero prime ideal is at most a single point, clarifying the one-point nature of prime tropical varieties, and establishes a dimension formula: the Krull-type dimension of the coordinate semiring ${\mathbb T}[x_1,\dots,x_n]/Bend(I)$ equals the tropical dimension of $\mathbb V(I)$ plus 1. The results connect algebraic and combinatorial aspects of tropical geometry, offering a framework to compute tropical dimensions via coordinate semirings and highlighting when tropical ideals are tropicalizations of classical data. An appendix strengthens the foundation by giving an alternative proof that prime tropical ideals correspond to tropicalizations of points.
Abstract
In this note we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its coordinate semiring.