Products of polymatroids with the strong exchange property

TL;DR

This paper extends the understanding of toric rings associated with polymatroids, demonstrating that products of polymatroids with the strong exchange property satisfy White's conjecture, broadening previous results.

Contribution

It generalizes prior results by proving White's conjecture for products of polymatroids with the strong exchange property.

Findings

Proves White's conjecture for product polymatroids with the strong exchange property.

Extends Conca's results on transversal polymatroids.

Shows that the toric ring is defined by symmetric exchange relations in this broader class.

Abstract

It was conjectured by White in 1980 that the toric ring associated to a matroid is defined by symmetric exchange relations. This conjecture was extended to discrete polymatroids by Herzog and Hibi, and they prove that the conjecture holds for polymatroids with the so called strong exchange property. In this paper we generalize their result to polymatroids that are products of polymatroids with the strong exchange property. This also extends a result by Conca on transversal polymatroids.