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arXiv · 2507.13153

Syzygies of polymatroidal ideals

Yairon Cid-Ruiz·Jacob P. Matherne·Anna Shapiro
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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.

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Yairon Cid-Ruiz, Jacob P. Matherne, Anna Shapiro. 2025-07-17. Syzygies of polymatroidal ideals. https://arxiv.org/abs/2507.13153

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