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arxiv: 2502.18396 · v2 · submitted 2025-02-25 · 🧮 math.AC · math.CO

Square-free powers of Cohen-Macaulay simplicial forests

Kanoy Kumar Das , Amit Roy , Kamalesh Saha This is my paper

Pith reviewed 2026-05-23 02:33 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords square-free powersCohen-Macaulay simplicial forestsfacet idealsdepth formulaspecial leavesnormalized depth function
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The pith

Cohen-Macaulay simplicial forests yield Cohen-Macaulay rings for every square-free power of the facet ideal.

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

The paper shows that when a simplicial complex is a Cohen-Macaulay forest, the polynomial ring modulo the k-th square-free power of its facet ideal remains Cohen-Macaulay for any positive integer k. This property fails for ordinary powers unless the ideal is a complete intersection. The authors define a special leaf to recursively compute an explicit formula for the depth of these quotient rings. They apply this to prove that the normalized depth function for such forests is nonincreasing.

Core claim

If Δ is a Cohen-Macaulay simplicial forest, then R/I(Δ)^[k] is Cohen-Macaulay for all k ≥ 1. The proof relies on introducing special leaves and providing a combinatorial formula for depth(R/I(Δ)^[k]) that is independent of the grading.

What carries the argument

The special leaf: a combinatorial feature allowing successive removal of vertices to produce an explicit non-negative integer formula for the depth of each square-free power.

If this is right

  • The normalized depth function of any Cohen-Macaulay simplicial forest is nonincreasing.
  • Explicit combinatorial formulas for depth become available for all square-free powers.
  • Square-free powers preserve the Cohen-Macaulay property in this class, unlike ordinary powers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The leaf-removal technique may extend to other classes of simplicial complexes that admit analogous recursive structures.
  • Links to matching theory in hypergraphs could produce new combinatorial invariants.
  • Direct computation on small explicit forests would allow immediate numerical checks of the depth formula.

Load-bearing premise

A Cohen-Macaulay simplicial forest always admits a sequence of special leaves whose successive removal yields the depth formula.

What would settle it

A counterexample would be any Cohen-Macaulay simplicial forest Δ together with an integer k such that depth(R/I(Δ)^[k]) is strictly smaller than the Krull dimension of the ring.

Figures

Figures reproduced from arXiv: 2502.18396 by Amit Roy, Kamalesh Saha, Kanoy Kumar Das.

Figure 1
Figure 1. Figure 1: FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIGURE 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

Let $I(\Delta)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $\Delta$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $\Delta$ is a Cohen-Macaulay simplicial forest, then $R/I(\Delta)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(\Delta)^{[k]})$ for all $k\ge 1$, where $\Delta$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.

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 proves that if Δ is a Cohen-Macaulay simplicial forest then R/I(Δ)^[k] is Cohen-Macaulay for every k ≥ 1. The argument proceeds by introducing the combinatorial notion of a special leaf, constructing an explicit formula for depth(R/I(Δ)^[k]) via successive removal of special leaves, and verifying that this depth equals the Krull dimension independently of k; an application shows that the normalized depth function is nonincreasing.

Significance. The result supplies a combinatorial depth formula that is independent of the algebraic grading and yields the Cohen-Macaulay property for all square-free powers, in contrast to the known behavior of ordinary powers of radical ideals. The explicit construction via special leaves and the verification that the resulting depth is always non-negative constitute a concrete, falsifiable contribution to the literature on facet ideals and square-free powers.

minor comments (3)
  1. [§2] §2 (definition of special leaf): the inductive removal argument would benefit from an explicit statement that the depth formula remains non-negative after each removal step, even if this is verified later in the proof.
  2. [Abstract] The abstract refers to 'Matching Theory' without a citation; a reference to the relevant matching-theoretic literature on square-free powers would clarify the context.
  3. [Application section] Notation for the normalized depth function is introduced only in the application section; moving its definition to the preliminaries would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained combinatorial construction

full rationale

The paper introduces the new combinatorial notion of a special leaf for Cohen-Macaulay simplicial forests and uses successive removal of such leaves to obtain an explicit, non-negative integer formula for depth(R/I(Δ)^[k]) that is shown to equal the Krull dimension independently of k. This is presented as a direct inductive argument on the combinatorial structure rather than a fitted parameter, self-referential definition, or load-bearing self-citation. No equations or steps reduce the claimed depth formula to its own inputs by construction, and the central claim remains independent of any prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger records the background notions invoked in the statement itself.

axioms (2)
  • standard math Facet ideal I(Δ) of a simplicial complex Δ is a monomial ideal generated by the products of variables corresponding to facets
    Standard definition in the field of combinatorial commutative algebra, invoked when the abstract refers to I(Δ)^[k]
  • standard math Cohen-Macaulay property for graded rings and for simplicial complexes
    Background notion from commutative algebra used to state both the hypothesis on Δ and the conclusion on R/I(Δ)^[k]
invented entities (1)
  • special leaf no independent evidence
    purpose: Combinatorial object used to give an explicit formula for depth(R/I(Δ)^[k])
    New notion introduced in the paper to prove the main result

pith-pipeline@v0.9.0 · 5723 in / 1587 out tokens · 42314 ms · 2026-05-23T02:33:14.729753+00:00 · methodology

discussion (0)

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Reference graph

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