Cave Polynomial in Polymatroids

• Cave Polynomial is a polynomial invariant of polymatroids that encodes combinatorial, geometric, K-theoretic, and homological information using lattice points and the Möbius function.
• It has equivalent combinatorial formulations—stalactite, box, and Möbius polynomials—that offer distinct algorithmic insights into its computation and underlying structure.
• Its valuative and duality properties bridge the cave polynomial with K-polynomials and syzygy data, enhancing the analysis of polymatroidal ideals in algebraic geometry.

The cave polynomial is a polynomial invariant of a polymatroid that was introduced to encode, in a directly combinatorial form, data that is simultaneously combinatorial, geometric, $K$-theoretic, and homological. For a polymatroid $P$, it is defined from the lattice points of the base polytope by a product formula that detects admissible one-step coordinate transfers, and its coefficients are exactly the values of the polymatroid Möbius function. The invariant is valuative, its support after homogenization is again a generalized polymatroid, and via duality it determines the $K$-polynomial and homological shift ideals of the associated polymatroidal ideal (Cid-Ruiz et al., 17 Jul 2025).

1. Definition and polymatroidal setting

A polymatroid on $[p]=\{1,\dots,p\}$ with cage $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$ may be described by a rank function

$\rho_P:2^{[p]}\to \mathbb N$

satisfying normalization, monotonicity, submodularity, and the cage condition

$$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$

Its rank is

$\operatorname{rk}(P)=\rho_P([p]).$

A polymatroid with cage $(1,\dots,1)$ is a matroid. The associated base and independence polytopes are

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$

and

$P$0

The identity

$P$1

is repeatedly used in the theory (Cid-Ruiz et al., 17 Jul 2025).

The same object can also be viewed as a finite homogeneous $P$2-convex subset $P$3. In that model, a finite set $P$4 is homogeneous if all points have the same coordinate sum, and $P$5 is $P$6-convex when for each $P$7 and $P$8 such that $P$9, there is $K$0 such that $K$1 and $K$2. In the point-set model,

$K$3

and $K$4 consists of lattice points “under” the base polytope (Shapiro, 12 Jan 2026).

In the notation of the 2025 paper, the cave polynomial is

$K$5

where $K$6, $K$7 is the $K$8-th standard basis vector, and $K$9 is the indicator of the base polytope. The authors emphasize that $[p]=\{1,\dots,p\}$0 is an honest polynomial in $[p]=\{1,\dots,p\}$1, not a Laurent polynomial. A lexicographically permuted version exists for any $[p]=\{1,\dots,p\}$2, and the resulting polynomial is independent of that choice (Cid-Ruiz et al., 17 Jul 2025).

The combinatorial meaning of the product is local: for a base point $[p]=\{1,\dots,p\}$3, the factor indexed by $[p]=\{1,\dots,p\}$4 records whether one can move one unit from coordinate $[p]=\{1,\dots,p\}$5 to some later coordinate $[p]=\{1,\dots,p\}$6 and remain in the base polytope. Expanding the product therefore produces lower-degree monomials beneath a top-degree base monomial. This local “downward” structure motivates the cave terminology.

2. Equivalent combinatorial formulas

A central structural result is that three apparently different formulas compute the same invariant: $[p]=\{1,\dots,p\}$7 The 2026 paper proves these equalities by direct combinatorial arguments and presents them as three algorithms for computing the cave polynomial (Shapiro, 12 Jan 2026).

The first formulation is the stalactite polynomial. For $[p]=\{1,\dots,p\}$8, $[p]=\{1,\dots,p\}$9 is a neighbor of $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$0 in the $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$1 direction if $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$2. Given distinct $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$3, the associated stalactite is

$\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$4

If $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$5, one defines $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$6 using the coordinates for which $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$7 has a neighbor in $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$8. With a lexicographic order $\mathbf m=(m_1,\dots,m_p)\in \mathbb N^p$9 on the base points, one forms the stalactites

$\rho_P:2^{[p]}\to \mathbb N$0

counts how many stalactites contain each $\rho_P:2^{[p]}\to \mathbb N$1, and sets

$\rho_P:2^{[p]}\to \mathbb N$2

where $\rho_P:2^{[p]}\to \mathbb N$3 is the number of stalactites containing $\rho_P:2^{[p]}\to \mathbb N$4. The equality $\rho_P:2^{[p]}\to \mathbb N$5 is termwise: choosing a subset of factors $\rho_P:2^{[p]}\to \mathbb N$6 in the cave formula is exactly choosing a subset of downward stalactite directions.

The second formulation is the box polynomial

$\rho_P:2^{[p]}\to \mathbb N$7

This expression sums over all lattice points in the independence polytope rather than over top-degree base points. For each $\rho_P:2^{[p]}\to \mathbb N$8, the product expands by subtracting at most one from each positive coordinate. The combinatorial content is an inclusion–exclusion expansion over points of the form $\rho_P:2^{[p]}\to \mathbb N$9.

The third formulation is the Möbius polynomial

$$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$0

defined from the poset $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$1 with adjoined maximal element $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$2, ordered componentwise. The interval Möbius function has an explicit formula: $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$3 This theorem identifies the nonzero intervals as exactly the Boolean ones obtained by lowering a set of distinct coordinates by one. It follows that the coefficient of $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$4 in the $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$5-summand of the box polynomial is precisely $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$6, from which $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$7 follows by Möbius inversion.

These equivalences have methodological significance. Earlier arguments relating the formulas used algebraic geometry and multiplicity-free varieties; the 2026 paper replaces those arguments with direct combinatorial proofs (Shapiro, 12 Jan 2026).

3. Möbius interpretation, support, and valuativity

The coefficient description of the cave polynomial is one of the principal results of the theory: $$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$8 Here the Möbius function is defined recursively by

$$\rho_P(\{i\})\le m_i \quad \text{for all } i\in [p].$$9

$\operatorname{rk}(P)=\rho_P([p]).$0

and $\operatorname{rk}(P)=\rho_P([p]).$1 otherwise. The Möbius support is

$\operatorname{rk}(P)=\rho_P([p]).$2

Because the cave polynomial is the generating function of $\operatorname{rk}(P)=\rho_P([p]).$3, its support is exactly $\operatorname{rk}(P)=\rho_P([p]).$4 (Cid-Ruiz et al., 17 Jul 2025).

The support has strong convexity structure. Theorem A of the 2025 paper states that the support of the cave polynomial is a generalized polymatroid. The relevant definition is by homogenization: a polynomial $\operatorname{rk}(P)=\rho_P([p]).$5 has support a generalized polymatroid if the support of

$\operatorname{rk}(P)=\rho_P([p]).$6

is a base discrete polymatroid. Thus the support of the cave polynomial is not an arbitrary subset of $\operatorname{rk}(P)=\rho_P([p]).$7; after homogenization it inherits precisely the discrete polymatroid structure needed in subsequent applications.

The same theorem states that the map

$\operatorname{rk}(P)=\rho_P([p]).$8

is valuative. The 2025 paper proves this first as Proposition 2.16, reducing it to valuativity of the multigraded Hilbert function of the associated polymatroidal ideal and using the fact that the valuative group is generated by indicator functions of realizable polymatroids over $\operatorname{rk}(P)=\rho_P([p]).$9. This valuativity is the device that transports geometric identities from realizable to arbitrary polymatroids (Cid-Ruiz et al., 17 Jul 2025).

The 2026 paper gives a purely combinatorial shadow of this Möbius interpretation. It proves that the stalactite counts satisfy the same recurrence: $(1,\dots,1)$0 and hence

$(1,\dots,1)$1

This identifies the coefficients simultaneously as stalactite counts and Möbius values (Shapiro, 12 Jan 2026).

4. $(1,\dots,1)$2-theoretic, geometric, and homological roles

The cave polynomial was introduced to study syzygies of polymatroidal ideals. For

$(1,\dots,1)$3

with standard $(1,\dots,1)$4-grading, the associated polymatroidal ideal is

$(1,\dots,1)$5

If

$(1,\dots,1)$6

is the minimal $(1,\dots,1)$7-graded free resolution of $(1,\dots,1)$8, then the $(1,\dots,1)$9-th homological shift ideal is

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$0

The $$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$1-polynomial is

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$2

Theorem A states that

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$3

where $$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$4 is the dual polymatroid. Consequently,

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$5

Thus the cave polynomial of the dual polymatroid determines all multigraded shifts in the resolution (Cid-Ruiz et al., 17 Jul 2025).

The same paper interprets the cave polynomial $$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$6-theoretically. With

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$7

the dual polymatroidal ideal

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$8

defines the polymatroidal multiprojective variety

$$B(P)=\left\{v\in \mathbb R_{\ge 0}^p \,\middle|\, \sum_{i=1}^{p}v_i=\operatorname{rk}(P)\ \text{ and }\ \sum_{j\in J} v_j \le \rho_P(J)\ \text{for all } J\subseteq [p]\right\},$$9

If

$P$00

then the coefficients of the cave polynomial satisfy

$P$01

The cave polynomial therefore records the structure sheaf class of $P$02 in the natural basis of the augmented $P$03-ring.

Its relation to the Snapper polynomial makes the same information accessible in binomial basis form. If $P$04 is a multisymmetric lift with partition $P$05, the Snapper polynomial is

$P$06

With the $P$07-linear map

$P$08

one has

$P$09

The 2026 paper restates this as

$P$10

showing that a geometric Euler characteristic invariant can be computed from Möbius inversion or from stalactite combinatorics (Shapiro, 12 Jan 2026).

These properties resolve two conjectures recorded in the 2025 paper. The support statement proves the conjecture of Castillo–Cid-Ruiz–Mohammadi–Montaño that the Möbius support of a polymatroid is a generalized polymatroid, and the syzygy formula proves the conjecture of Bandari–Bayati–Herzog that homological shift ideals of polymatroidal ideals are again polymatroidal (Cid-Ruiz et al., 17 Jul 2025).

5. Computations and examples

A small example from the 2026 paper illustrates the equality of all four formulas. Let

$P$11

Then

$P$12

and the independence lattice points are

$P$13

With lex order

$P$14

the stalactites are

$P$15

$P$16

$P$17

Hence

$P$18

The cave formula gives exactly the same polynomial: $P$19 The Möbius values are

$P$20

$P$21

and

$P$22

so

$P$23

The box formula also reduces to the same expression (Shapiro, 12 Jan 2026).

A larger example in the 2025 paper exhibits the interaction with duality and syzygies. For the polymatroid on $P$24 with cage $P$25 and rank function

$P$26

$P$27

the base lattice points are

$P$28

and

$P$29

Its dual $P$30 has base points

$P$31

and cave polynomial

$P$32

The relation

$P$33

then recovers the $P$34-polynomial and hence the syzygetic data of $P$35 (Cid-Ruiz et al., 17 Jul 2025).

6. Terminological scope and related named polynomials

The cave polynomial belongs to the theory of polymatroids and polymatroidal ideals, and it should be distinguished from similarly named classical objects. The Cantor packing polynomials

$P$36

are polynomial bijections $P$37, and the Fueter–Pólya theorem states that every quadratic packing polynomial is one of these two (Nathanson, 2015). The Cayley cubic polynomial

$P$38

defines the Cayley surface and is studied through its integer points and Chebyshev- and Lucas-type parameterizations (Son, 2021). Neither object is the cave polynomial.

Within its own domain, the cave polynomial is best understood as a unifying invariant. It packages the Möbius function of a polymatroid, the structure sheaf class of a polymatroidal multiprojective variety, the binomial-basis form of the Snapper polynomial, and the support data governing the syzygies of a polymatroidal ideal. The 2025 paper introduces this framework and proves its major algebraic consequences, while the 2026 paper provides a combinatorial synthesis showing that caves and stalactites, box inclusion–exclusion, and Möbius inversion are three exact realizations of the same polynomial invariant (Cid-Ruiz et al., 17 Jul 2025).