Syzygies of polymatroidal ideals

TL;DR

This paper introduces the cave polynomial for polymatroids, providing a new valuative function and K-theoretic description, and resolves two existing conjectures related to polymatroidal ideals and M"obius support.

Contribution

It defines the cave polynomial for polymatroids, demonstrating its properties and applications, including settling two significant conjectures in the field.

Findings

The cave polynomial is a valuative function on polymatroids.

Homogenization of the cave polynomial support yields a polymatroid.

The paper resolves two conjectures in polymatroid theory.

Abstract

We introduce the cave polynomial of a polymatroid and show that it yields a valuative function on polymatroids. The support of this polynomial after homogenization is again a polymatroid. The cave polynomial gives a $K$-theoretic description of a polymatroid in the augmented $K$-ring of a multisymmetric lift. As applications, we settle two conjectures: one by Bandari, Bayati, and Herzog regarding polymatroidal ideals, and another by Castillo, Cid-Ruiz, Mohammadi, and Monta\~no regarding the M\"obius support of a polymatroid.