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arxiv: 2606.00599 · v1 · pith:2FN7VPVLnew · submitted 2026-05-30 · 🧮 math.CO

Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices

Yanru Chen , Houshan Fu , Ang Li , Suijie Wang This is my paper

Pith reviewed 2026-06-28 18:41 UTC · model grok-4.3

classification 🧮 math.CO
keywords convolution formulacharacteristic polynomialgeometric semilatticecentralizationlocalizationfinite field methodhyperplane arrangements
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The pith

A convolution formula holds for the characteristic polynomial of any finite geometric semilattice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the characteristic polynomial χ of a finite geometric semilattice M at argument st equals a sum over the centralization of M. Each term in the sum weights the product of the characteristic polynomial of a restricted centralization by that of the localization at the element X. This generalizes an earlier identity recovered by setting the parameter s to one. Specialization to hyperplane arrangements produces a new expansion of the polynomial.

Core claim

We establish the identity χ(M,st)=∑_{X∈underline{M}} s^{r-rk_underline{M}(X)} χ(underline{M}^X,t) χ(M_{(X)},s) for the characteristic polynomial of a finite geometric semilattice M. The proof uses a finite-field counting argument over the fields F_{p^2} and F_p to give a combinatorial interpretation. When applied to hyperplane arrangements the identity gives a new expansion related to Wang's formula.

What carries the argument

The convolution formula that decomposes the characteristic polynomial using centralization and localization operations on the semilattice.

Load-bearing premise

The semilattice must be finite and geometric so that centralization and localization are defined and the counting argument applies.

What would settle it

Direct computation of both sides of the proposed identity for a small finite geometric semilattice, such as the Boolean lattice of rank 3, to check if they agree for specific values of s and t.

read the original abstract

We establish a convolution formula for the characteristic polynomial of a finite geometric semilattice $M$: \[ \chi(M,st)=\sum_{X\in \underline{M}} s^{r-{\rm rk}_{\underline{M}}(X)}\chi(\underline{M}^X,t)\,\chi(M_{(X)},s), \] where $\underline{M}$ denotes the centralization of $M$, and $M_{(X)}$ denotes the localization at $X$. This generalizes a nice formula of Southerland, Southern, and Zhou, which is recovered at $s=1$. When specialized to hyperplane arrangements, the identity yields a new expansion closely related to Wang's convolution formula. We further provide a combinatorial interpretation of the convolution formula using the finite field method over $\mathbb{F}_{p^2}$ and $\mathbb{F}_p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper establishes a convolution formula for the characteristic polynomial of a finite geometric semilattice M: χ(M,st)=∑_{X∈M̲} s^{r-rk_M̲(X)} χ(M̲^X,t) χ(M_{(X)},s), where M̲ denotes the centralization of M and M_{(X)} the localization at X. This generalizes the s=1 case of Southerland-Southern-Zhou and yields a new expansion for hyperplane arrangements; the proof is via a combinatorial finite-field counting argument over F_{p^2} and F_p.

Significance. If the identity holds, it supplies a new structural relation among characteristic polynomials on geometric semilattices that recovers known special cases and admits a direct combinatorial interpretation. The finite-field method is a standard, parameter-free tool in this area and constitutes a clear strength of the argument.

minor comments (3)
  1. The abstract and introduction should explicitly recall the definitions of centralization M̲ and localization M_{(X)} (including the rank function rk_M̲) before stating the main identity, to make the formula self-contained for readers outside the immediate subfield.
  2. In the finite-field counting argument, the precise choice of the two fields F_{p^2} and F_p and the role of the parameter s should be spelled out with a short diagram or table showing which counting is performed in which field.
  3. The manuscript should include a brief comparison (perhaps in a remark) of the new expansion obtained for hyperplane arrangements with Wang’s convolution formula, indicating exactly which terms coincide and which are new.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the accurate summary of the main result and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central result is a convolution identity for the characteristic polynomial of finite geometric semilattices, proved via the standard finite-field counting method over F_{p^2} and F_p. This is an independent combinatorial argument that does not reduce to any fitted input, self-definition, or load-bearing self-citation. The formula is stated to generalize a known case at s=1 and to recover a related expansion for arrangements, but these are explicit extensions rather than circular reductions. The modeling premise (finite geometric semilattice) is required for the operations to be defined and is not smuggled in via citation. No step in the provided abstract or description exhibits the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definitions and properties of finite geometric semilattices, centralization, and localization; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Finite geometric semilattices admit well-defined centralization and localization operations under which the characteristic polynomial is defined and satisfies the stated convolution.
    Invoked when the formula is stated for M.

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Reference graph

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19 extracted references · 3 canonical work pages · 1 internal anchor

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