Syzygies of polymatroidal ideals
Commutative Algebra
2025-07-18 v1 Algebraic Geometry
Combinatorics
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 -theoretic description of a polymatroid in the augmented -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.
Keywords
Cite
@article{arxiv.2507.13153,
title = {Syzygies of polymatroidal ideals},
author = {Yairon Cid-Ruiz and Jacob P. Matherne and Anna Shapiro},
journal= {arXiv preprint arXiv:2507.13153},
year = {2025}
}