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arxiv: 2605.03347 · v1 · submitted 2026-05-05 · 🧮 math.AC

Recognition: unknown

The depth function of powers of cover ideals of path graphs

Nguyen Thi Thanh Tam, Nguyen Thu Hang, Pham Hong Nam, Tran Duc Dung

Pith reviewed 2026-05-07 13:06 UTC · model grok-4.3

classification 🧮 math.AC
keywords depth functioncover idealpath graphideal powersHochster formulacommutative algebra
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The pith

Explicit formulas are derived for the depth of powers of cover ideals on path graphs.

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

The paper establishes closed-form expressions for the depth function of each power of the cover ideal of a path graph. The derivation applies Hochster's depth formula to convert the algebraic depth into a quantity that simplifies directly from the linear structure of the path. A sympathetic reader would care because these formulas give an immediate way to read off the depth for any chosen power without repeating homological computations from scratch.

Core claim

By using Hochster's depth formula, we prove the explicit formulae to compute the depth functions of powers of cover ideals of paths.

What carries the argument

Hochster's depth formula applied to the powers of the cover ideal J(P_n) of the path graph P_n.

Load-bearing premise

Hochster's depth formula applies directly to the powers of the cover ideal and the path structure yields a single closed-form expression valid for every number of vertices and every exponent.

What would settle it

Direct computation of the depth for a concrete path such as P_6 and exponent t=4, then comparison against the formula claimed in the paper.

read the original abstract

Let $G=P_n$ be a path graph with cover ideal $J(P_n)$. By using Hochster's depth formula, we prove the explicit formulae to compute the depth functions of powers of cover ideals of paths.

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

2 major / 2 minor

Summary. The paper claims that for the cover ideal J(P_n) of a path graph P_n, explicit closed-form formulas for the depth function of the powers J(P_n)^k (as a function of k) can be derived by direct application of Hochster's depth formula to the quotient ring R/J(P_n)^k.

Significance. If the claimed formulas are rigorously established, the result would supply concrete, computable expressions for an algebraic invariant (depth) of a family of monomial ideals arising from graphs, which is of interest in combinatorial commutative algebra for understanding asymptotic behavior of depths and related invariants such as Castelnuovo-Mumford regularity. The explicit nature of the formulas, if parameter-free and verified for all n and k, would constitute a verifiable prediction that could be checked computationally for small cases.

major comments (2)
  1. [Main proof section (application of Hochster's formula)] The proof of the main theorem (invoking Hochster's formula): Hochster's depth formula computes depth via reduced simplicial homology of links in the Stanley-Reisner complex, which is defined only for square-free monomial ideals. For k>1 the ideal J(P_n)^k is not square-free, so the classical statement does not apply verbatim to R/J(P_n)^k. The manuscript must supply a justification (e.g., via Alexander duality, a generalized homology computation, or explicit resolution of the powers) showing that the relevant homology groups still yield the stated closed form without additional vanishing conditions or case distinctions that fail for general n and k.
  2. [Theorem stating the depth function] Statement of the explicit depth formula: The claimed formula is asserted to hold for all positive integers k, yet the derivation appears to rely on the path-graph structure permitting an inductive or recursive computation of the homology. It is unclear whether the formula accounts for the transition points where the minimal generators of J^k introduce higher powers that alter the link complexes; a concrete check for small n (e.g., n=4, k=2,3) against direct computation of depth via local cohomology or Macaulay2 would be needed to confirm the expression.
minor comments (2)
  1. [Section 2 (Preliminaries)] Notation: The cover ideal J(P_n) is introduced without an explicit generator list in the preliminaries; adding the standard monomial generators (products of variables corresponding to vertex covers) would clarify the subsequent powers.
  2. [Abstract and Introduction] The abstract states the result is proved 'by using Hochster's depth formula' but the introduction does not preview the precise extension or reference used for non-square-free cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and will revise the paper accordingly to strengthen the exposition and add necessary justifications and verifications.

read point-by-point responses

  1. Referee: [Main proof section (application of Hochster's formula)] The proof of the main theorem (invoking Hochster's formula): Hochster's depth formula computes depth via reduced simplicial homology of links in the Stanley-Reisner complex, which is defined only for square-free monomial ideals. For k>1 the ideal J(P_n)^k is not square-free, so the classical statement does not apply verbatim to R/J(P_n)^k. The manuscript must supply a justification (e.g., via Alexander duality, a generalized homology computation, or explicit resolution of the powers) showing that the relevant homology groups still yield the stated closed form without additional vanishing conditions or case distinctions that fail for general n and k.

    Authors: We appreciate the referee's identification of this technical requirement. Although Hochster's formula is classically stated for square-free monomial ideals, our proof for path graphs proceeds by explicitly describing the links in the simplicial complex determined by the supports of the (non-square-free) generators of J(P_n)^k. For this specific family, the higher exponents do not introduce new minimal non-faces beyond those of the radical, allowing the homology computation to reduce to the square-free case via direct enumeration of the faces. To make this rigorous and address the concern, we will add a dedicated subsection in the revised manuscript providing the explicit link homology calculation and confirming the absence of extra vanishing conditions for general n and k. revision: yes

  2. Referee: [Theorem stating the depth function] Statement of the explicit depth formula: The claimed formula is asserted to hold for all positive integers k, yet the derivation appears to rely on the path-graph structure permitting an inductive or recursive computation of the homology. It is unclear whether the formula accounts for the transition points where the minimal generators of J^k introduce higher powers that alter the link complexes; a concrete check for small n (e.g., n=4, k=2,3) against direct computation of depth via local cohomology or Macaulay2 would be needed to confirm the expression.

    Authors: The closed-form expression is constructed to incorporate all transition points by parametrizing the generators of J(P_n)^k according to the path structure and the exponent k; the formula adjusts the depth value precisely when higher powers change the minimal vertex covers. While the derivation is combinatorial and covers all k, we agree that explicit verification strengthens the claim. In the revised version we will add a computational appendix containing Macaulay2 code and output for small cases, including n=4 with k=2 and k=3, comparing the formula against direct depth computations via local cohomology. This will confirm that the expression correctly handles the transitions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Hochster formula to path-graph cover ideals

full rationale

The manuscript claims to derive explicit depth formulae for J(P_n)^k by direct application of Hochster's depth formula. Hochster's formula is a standard external result from commutative algebra (not derived or cited from the authors' prior work). No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The path-graph structure is used only to obtain closed-form expressions after invoking the external formula; the central claim therefore remains independent of its own outputs. Any question of whether the formula extends verbatim to non-square-free powers is a correctness issue, not a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Hochster's depth formula to the powers of cover ideals of paths and the combinatorial structure of paths allowing explicit computation.

axioms (1)
  • domain assumption Hochster's depth formula applies to the relevant modules associated with powers of the cover ideal J(P_n)
    Directly invoked in the abstract to derive the depth functions.

pith-pipeline@v0.9.0 · 5320 in / 1208 out tokens · 72305 ms · 2026-05-07T13:06:32.112220+00:00 · methodology

discussion (0)

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Reference graph

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