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arxiv: 2411.10209 · v2 · submitted 2024-11-15 · 🧮 math.AC · math.CO

Gr\"obner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra

Filip Jonsson Kling , Samuel Lundqvist , Fatemeh Mohammadi , Matthias Orth , Eduardo S\'aenz-de-Cabez\'on This is my paper

Pith reviewed 2026-05-23 17:48 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords Gröbner basesweak Lefschetz propertystrong Lefschetz propertysquarefree algebraalmost complete intersectionslattice pathsCatalan numbersMayer-Vietoris trees
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The pith

Reduced Gröbner bases of ideals (x₁²,…,xₙ²,(∑x_i)^k) are indexed by lattice paths and classify the weak Lefschetz property.

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

The paper computes the reduced Gröbner basis for any term ordering of the almost complete intersection ideals generated by the squares of n variables and the kth power of their sum in the squarefree algebra. The basis elements receive a combinatorial indexing by lattice paths whose coefficients involve elementary symmetric polynomials and Catalan numbers. This explicit structure is then used to classify the weak Lefschetz property of the quotient rings. The same techniques supply a new proof that the squarefree algebra satisfies the strong Lefschetz property and permit construction of minimal free resolutions for the initial ideals by means of Mayer-Vietoris trees.

Core claim

For the ideals I = (x₁², …, xₙ², (x₁ + ⋯ + xₙ)^k) the reduced Gröbner basis with respect to any term order consists of elements whose leading terms are indexed by lattice paths, with coefficients drawn from elementary symmetric polynomials whose degrees relate to Catalan numbers. This combinatorial model classifies exactly for which parameters the quotient has the weak Lefschetz property, yields a new proof of the strong Lefschetz property for the squarefree algebra, and determines the Betti numbers of the initial ideals via Mayer-Vietoris trees.

What carries the argument

Reduced Gröbner basis whose monomials are indexed by lattice paths together with elementary symmetric polynomial coefficients.

If this is right

  • The weak Lefschetz property for these ideals is decided by conditions on n and k that arise from the lattice-path counts.
  • Betti numbers of the initial ideals are given explicitly by the same combinatorial data.
  • The squarefree algebra satisfies the strong Lefschetz property by a new argument that uses the Gröbner-basis structure.
  • Minimal free resolutions of the initial ideals can be constructed uniformly via Mayer-Vietoris trees.

Where Pith is reading between the lines

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

  • The lattice-path indexing may extend to Gröbner bases of other almost complete intersection ideals generated by squares and linear forms.
  • The appearance of Catalan numbers suggests possible links to known enumerative results in the study of Lefschetz properties or Hilbert series.
  • An explicit basis description could allow direct computation of further invariants such as Hilbert functions without running a Gröbner-basis algorithm.
  • The classification may inform analogous questions for Lefschetz properties in non-squarefree graded algebras or in positive characteristic.

Load-bearing premise

The combinatorial description of the reduced Gröbner basis via lattice paths holds for every term ordering and every positive integer k without restrictions on the base field or its characteristic.

What would settle it

For n=3 and k=2 compute the reduced Gröbner basis explicitly in any term order and verify whether the leading monomials match the lattice-path prediction and whether the resulting weak Lefschetz property classification agrees with direct computation of the relevant multiplication maps.

Figures

Figures reproduced from arXiv: 2411.10209 by Eduardo S\'aenz-de-Cabez\'on, Fatemeh Mohammadi, Filip Jonsson Kling, Matthias Orth, Samuel Lundqvist.

Figure 1
Figure 1. Figure 1: The (N, E)-lattice path associated to x1x3x4. However, Gn,k is in general not a universal Gr¨obner basis: For instance, in G5,2 (see Example 2.24), the polynomials g{1,3,4},5,2 and g{2,3,4},5,2 share the same leading term, x3x4x5, with respect to the degree reverse lexicographic ordering with x5 ≻ x4 ≻ x3 ≻ x2 ≻ x1. We aim to compute the number of different reduced Gr¨obner bases of the ideal In,k. We writ… view at source ↗
Figure 2
Figure 2. Figure 2: Kantor [19], Graver and Jurkat [11], and Wilson [37], with these references initially brought to their attention by Bernt Lindstr¨om. Besides the results from the 1970s, we are aware of the proof by Hara and Watanabe for the case d = 2, an argument by Ikeda [18] from the 1990s, and a more recent proof by Phuong and Tran [28]. Therefore, to the best of our knowledge, our proof represents the seventh contrib… view at source ↗
Figure 3
Figure 3. Figure 3: Partial display of the Mayer-Vietoris tree MVT(in(I4,2)). Proposition 4.7 ([33], Proposition 3). Let µ ∈ N n be a multidegree such that x µ is a minimal generator in exactly k relevant nodes of the Mayer-Vietoris tree of I, and all of these nodes have the same dimension i. Then, β(i,µ)(I) = k. In particular, if x µ appears as a minimal generator in only one relevant node, then β(i,µ)(I) = 1. Remark 4.8. In… view at source ↗
read the original abstract

For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gr\"obner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach.

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 / 0 minor

Summary. The manuscript computes the reduced Gröbner basis of the almost complete intersection ideals (x₁²,…,xₙ²,(x₁+⋯+xₙ)^k) for arbitrary term orderings, exhibiting an explicit combinatorial indexing by lattice paths that involves elementary symmetric polynomials and Catalan numbers. This structure is used to classify the weak Lefschetz property for these ideals. The paper also supplies a new proof that the squarefree algebra satisfies the strong Lefschetz property and constructs minimal free resolutions of the initial ideals via a Mayer-Vietoris tree, from which the Betti numbers are obtained.

Significance. If the claimed term-order-independent Gröbner bases and the lattice-path classification hold, the work supplies a concrete combinatorial tool for studying Lefschetz properties on a family of almost complete intersections that appears frequently in the literature. The uniform treatment across all term orders and the explicit Mayer-Vietoris resolution for the initial ideals are concrete strengths that would be of direct use to researchers working on monomial ideals and Lefschetz properties in squarefree algebras.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive summary, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central results are explicit computations: the reduced Gröbner basis of the almost complete intersection ideals (x₁²,…,xₙ²,(x₁+⋯+xₙ)^k) for arbitrary term orders, with a direct combinatorial indexing by lattice paths, elementary symmetric polynomials, and Catalan numbers. The weak Lefschetz classification follows immediately from this basis. A new proof is supplied for the known strong Lefschetz property of the squarefree algebra, and Betti numbers are obtained via an independent Mayer-Vietoris tree construction. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts about polynomial rings, Gröbner bases, and free resolutions; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of polynomial rings over a field and the existence of reduced Gröbner bases for any term ordering
    Invoked to guarantee that the reduced Gröbner basis exists and can be computed for arbitrary term orders.
  • domain assumption The linear form x₁ + ⋯ + xₙ is general
    The ideal is defined using this specific linear form; the combinatorial structure is claimed to hold for it.

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Works this paper leans on

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