Dilworth Truncations and Hadamard Products of Linear Spaces

Abstract.

As a direct application of Dilworth truncations of polymatroids, we give short proofs of two theorems: Bernstein’s characterization 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 characterization of the algebraic matroids of Hadamard products of more than two linear spaces, by giving explicit counterexamples.