Dilworth truncations and Hadamard products of linear spaces

TL;DR

This paper explores the properties of Dilworth truncations and Hadamard products of linear spaces, providing new proofs of existing theorems, and disproving a conjecture about algebraic matroids in this context.

Contribution

It offers concise proofs of key theorems related to algebraic matroids and Hadamard products, and presents counterexamples to Bernstein's conjecture.

Findings

Short proofs of Bernstein's theorem and a dimension formula for amoebas.

Disproof of Bernstein's conjecture for more than two linear spaces.

Explicit counterexamples to Bernstein's conjecture.

Abstract

As a direct application of Dilworth truncations of polymatroids, we give short proofs of two theorems: Bernstein's characterisation of algebraic matroids coming from the Hadamard product of two linear spaces, and a formula for the dimension of the amoeba of a complex linear space by Draisma, Eggleston, Pendavingh, Rau, and Yuen. We disprove Bernstein's conjecture on a characterisation of the algebraic matroids of Hadamard products of more than two linear spaces, by giving explicit counterexamples.