Algebraic Matroids and Set-theoretic Realizability of Tropical Varieties

TL;DR

This paper explores the relationship between algebraic matroids and tropical varieties, establishing conditions for realizability and demonstrating the existence of non-realizable Bergman fans.

Contribution

It introduces a necessary condition for tropical varieties to be realizable from prime ideals and shows infinitely many Bergman fans are not realizable.

Findings

Algebraic matroids are preserved under tropicalization.

A necessary condition for set-theoretic realizability of tropical varieties.

Existence of infinitely many non-realizable Bergman fans.

Abstract

To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.