Tropical bases by regular projections

TL;DR

This paper improves the understanding of tropical varieties by showing that prime ideals have short tropical bases with bounded size, and provides a computational method to find these bases.

Contribution

It introduces a bound on the size of tropical bases for prime ideals and offers a computational approach to construct them.

Findings

Prime ideals have tropical bases of size at most r + codim I + 1.

The degree of basis elements may increase, but the basis remains short.

A computational method for constructing these bases is provided.

Abstract

We consider the tropical variety $\mathcal{T}(I)$ of a prime ideal $I$ generated by the polynomials $f_1, ..., f_r$ and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that $I$ has a short tropical basis of cardinality at most $r + \codim I + 1$ at the price of increased degrees, and we provide a computational description of these bases.