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Three Combinatorial Algorithms for the Cave Polynomial of a Polymatroid

Anna Shapiro

TL;DR

This work establishes a unified combinatorial framework for the cave polynomial of a polymatroid by proving that three distinct polynomial constructions—cave, Stalactites, and Box—coincide with the Möbius polynomial in full generality. It provides explicit formulas for the underlying Möbius function and a Möbius-like recurrence for stalactites, yielding a purely combinatorial route to the equalities previously shown via algebraic geometry. The results also connect to the Snapper polynomial, showing its expressions align with these same combinatorial invariants, thereby linking multidegree data, Hilbert coefficients, and Grothendieck classes for both realizable and arbitrary polymatroids. The findings offer a transparent bridge between combinatorial, geometric, and representation-theoretic perspectives on polymatroids and their associated varieties. Overall, the paper clarifies how the cavity-like structure of a polymatroid encodes rich information about its algebraic and geometric invariants through a single, computable framework.

Abstract

The cave polynomial of a polymatroid was recently introduced and used to study the syzygies of polymatroidal ideals. We study the combinatorial relationships between three formulas for the cave polynomial. As an application, we interpret the Snapper polynomial in terms of these three formulas.

Three Combinatorial Algorithms for the Cave Polynomial of a Polymatroid

TL;DR

This work establishes a unified combinatorial framework for the cave polynomial of a polymatroid by proving that three distinct polynomial constructions—cave, Stalactites, and Box—coincide with the Möbius polynomial in full generality. It provides explicit formulas for the underlying Möbius function and a Möbius-like recurrence for stalactites, yielding a purely combinatorial route to the equalities previously shown via algebraic geometry. The results also connect to the Snapper polynomial, showing its expressions align with these same combinatorial invariants, thereby linking multidegree data, Hilbert coefficients, and Grothendieck classes for both realizable and arbitrary polymatroids. The findings offer a transparent bridge between combinatorial, geometric, and representation-theoretic perspectives on polymatroids and their associated varieties. Overall, the paper clarifies how the cavity-like structure of a polymatroid encodes rich information about its algebraic and geometric invariants through a single, computable framework.

Abstract

The cave polynomial of a polymatroid was recently introduced and used to study the syzygies of polymatroidal ideals. We study the combinatorial relationships between three formulas for the cave polynomial. As an application, we interpret the Snapper polynomial in terms of these three formulas.
Paper Structure (7 sections, 11 theorems, 33 equations)

This paper contains 7 sections, 11 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The realizable case
  3. The arbitrary case
  4. Combinatorial Approach
  5. Preliminaries
  6. Formula for the Möbius Function of a polymatroid
  7. Stalactites are counted by a Möbius function

Key Result

Theorem 1.1

The three polynomials above are all equal to the cave polynomial. That is, for any polymatroid $\mathscr{P}$, we have

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Corollary 1
  • Definition 5
  • ...and 23 more