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The Gr\"obner basis for powers of a general linear form in a monomial complete intersection

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Pith's one-line read This paper gives an explicit description of the reduced Gröbner basis of every ideal generated by powers of the variables together with a power of their sum.

desk verdict A likely-correct main theorem with an explicit, checkable construction, but the reducedness proof rests on a lemma that is false as stated; the gap is real and repairable. read the letter →

arxiv 2506.24028 v1 pith:E3UXSURV submitted 2025-06-30 math.AC math.COmath.RA

Filip Jonsson Kling , Samuel Lundqvist , Fatemeh Mohammadi , Matthias Orth This is my paper · ORCID
classification math.ACmath.COmath.RA MSC 13P1013E1005A1513D40
keywords GröbnerbasismonomialcompleteintersectionalmostinitialideallatticepathsreflectionmapLefschetzpropertiesCatalannumbers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the almost complete intersection ideals $I_{n,m,k} = (x_1^{m_1}, \ldots, x_n^{m_n}, (x_1+\cdots+x_n)^k)$ and claims to determine their reduced Gröbner bases completely, for any choice of exponents and any term order whose variable ranking is $x_1 \succ \cdots \succ x_n$. The description is explicit: each basis element is a pure power $x_j^{m_j}$ together with a polynomial $g_s$ whose leading monomial is indexed by a critical lattice path, and whose coefficients are explicit products of binomial coefficients. If the description is correct, it provides the first closed-form Gröbner basis for arbitrary powers of the linear form in this setting, and it turns the initial ideals into a combinatorial object that can be counted and compared by degree. The authors use the resulting Hilbert-series identity to reprove the strong Lefschetz property for Artinian monomial complete intersections in characteristic zero, and to connect degree sequences of the Gröbner bases to Catalan, Motzkin, Riordan, and spin-$s$-Catalan numbers.

What carries the argument

The key machinery is a lattice-path model of the $m$-free monomials in $R/P_{n,m}$: each monomial $t=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ maps to a path whose $i$-th step has slope $1-\alpha_i$ (up, flat, or down), and a piecewise-linear red curve $L_{n,m,k}$ is drawn through column midpoints shifted down by $k/2$. A monomial is critical if reflecting the tail of its path across $L_{n,m,k}$ produces another admissible path; the reflection map $\Lambda_d$ pairs degree $d$ with degree $d' = \sum_i m_i - n - d + k$ and is a bijection (Lemma 3.13), which yields the Hilbert-series truncation. The Gröbner basis polynomials are then produced by a binomial-coefficient identity (Lemma 4.1 and Proposition 4.3) showing $g_s = f_s \cdot \ell^k$ in the quotient ring, so each critical monomial is the leading term of an element of $I_{n,m,k}$.

What would settle it

Check the reduced Gröbner basis of $I_{2,(2,2),2} = (x_1^2,x_2^2,(x_1+x_2)^2)$: for the critical monomial $s=x_1x_2$, the raw polynomial $g_s$ has tail $\tfrac12 x_2^2$, which lies in $\operatorname{in}(I)$, so Lemma 4.7 as stated is false and the theorem's reduction clause is what removes this tail; a convincing falsification would be any parameter triple $(n,m,k)$ and critical $s$ whose reduced expansion still contains a monomial divisible by $x_j^{m_j}$, which would contradict Theorem 1.1's reducedness claim.

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Extended reading notes

Core claim

The discovery is a structural theorem: with a term order compatible with $x_1 \succ \cdots \succ x_n$, the initial ideal of $I_{n,m,k}$ is exactly the monomial ideal generated by $x_1^{m_1}, \ldots, x_n^{m_n}$ and the monomials whose lattice paths are critical under a reflection rule across a piecewise-linear curve $L_{n,m,k}$. Theorem 3.19 gives the minimal generators explicitly, Theorem 4.5 identifies this monomial ideal as $\operatorname{in}(I_{n,m,k})$, and Theorem 4.8 (restated as Theorem 1.1) writes each Gröbner basis polynomial as $g_s = \sum_{s'' \mid s'} \lambda_{s''} s'' (x_j+\cdots+x_n)^{\deg(s)-\deg(s'')}$ with binomial coefficients, reduced modulo the variable powers when needed. The numerical heart of the identification is the Hilbert-series equality $\operatorname{HS}(R/(M_{n,m,k});t) = [(1-t^k)\operatorname{HS}(R/P_{n,m};t)]$, which comes from the reflection pairing of critical paths and drives the Lefschetz applications.

Load-bearing premise

The load-bearing step is Lemma 4.7's claim that every non-leading monomial of each explicitly written polynomial $g_s$ lies outside the initial ideal; the reducedness of the proposed basis collapses if, for some critical monomial $s$, a tail term of $g_s$ belongs to that ideal after the reduction step.

Editorial extensions

If this is right

  • Every ideal $I_{n,m,k}$ now has a closed-form reduced Gröbner basis for any term order respecting $x_1 \succ \cdots \succ x_n$, removing the need to run Buchberger's algorithm for individual choices of $n,m,k$.
  • The initial ideals of $I_{n,m,k}$ are strongly $m$-stable (Corollary 3.23), resolving a stability question posed for the squarefree case and giving a uniform combinatorial description of their minimal generators.
  • Artinian monomial complete intersections over characteristic zero obtain a new proof of the strong Lefschetz property, via the Hilbert-series truncation equality instead of the classical Stanley-Watanabe argument.
  • In the equigenerated case, the number of Gröbner basis elements in each degree is exactly a convolution of Motzkin and Riordan numbers for $m=3$ and a shift of Catalan convolutions for $m=2$; for $k=1$ these counts are spin-$s$-Catalan numbers, and the degeneracy of the $W_0$ eigenvalue of an entanglement witness is $g_{m,1}(\sigma N+1)$.
  • For $n \geq 5$ and $k=1$, the weak Lefschetz property of $R/P_{n,m}$ in characteristic $p$ is equivalent to the initial ideal of $I_{n,m,1}$ coinciding with its characteristic-zero initial ideal (Theorem 5.14).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same reflection pairing suggests a direct bijective proof that the rows of the $(m-1)$-Catalan triangle are log-concave, and the lattice-path decomposition in Theorem 5.5 may extend to arbitrary $m$ to give closed forms for the sequences $g_{m,k}$ as products of generating functions indexed by blocks of size $m-1$.
  • Editorial inference: Proposition 5.13 could be turned into a practical sufficient test for the weak Lefschetz property in mixed degrees: check whether any prime $p$ divides a leading coefficient of $G_{n,m,1}$; Example 5.16 shows the converse can fail, so the test would not be necessary.
  • Editorial inference: because $G_{n,m,k}$ depends only on the variable ordering, the bound of $n!$ distinct reduced Gröbner bases (tight for equigenerated $m \geq 3$) gives a way to count initial ideals of these almost complete intersections and to track how Betti numbers vary across the Gröbner fan.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the ideals I_{n,m,k} = (x_1^{m_1},...,x_n^{m_n},(x_1+...+x_n)^k) in a polynomial ring over a field of characteristic zero. The authors develop a lattice-path model for m-free monomials and a reflection operation, and use it to identify the initial ideal in(I_{n,m,k}) with an explicitly described monomial ideal (M_{n,m,k}). Their main theorem (Theorem 1.1/4.8) gives an explicit reduced Gröbner basis for any monomial ordering with x_1 ≻ ... ≻ x_n, consisting of the pure powers together with polynomials g_s associated to certain critical monomials s. As applications, they give a new proof of the strong Lefschetz property for monomial complete intersections, connections of the Gröbner basis degree sequences to Catalan, Motzkin, and Riordan numbers, and a characteristic-p criterion for the weak Lefschetz property. The paper is ambitious and self-contained, with the main proof built on a series of combinatorial lemmas.

Significance. If the main theorem is correct, it would be the first complete explicit description of the reduced Gröbner bases for this natural family of almost complete intersection ideals, unifying and extending the special cases previously treated in the literature. The combinatorial machinery is elegant, and the derived connections to Catalan, Motzkin, and Riordan numbers, as well as to the strong Lefschetz property and to entanglement witnesses in quantum physics, are interesting and potentially impactful. The paper does not rely on fitted parameters or circular reasoning; the reflection map is defined from the input data, and the Hilbert-series comparisons are standard. However, the reducedness of the claimed basis is not established as written because of a false lemma, and this gap affects the central theorem and several downstream results.

major comments (2)
  1. [§4.1, Lemma 4.7] Lemma 4.7 is false as stated. For n=2, m=(2,2), k=2, and s=x_1x_2, Setup 4.2(i) gives g_s = x_1x_2 + (1/2)x_2^2. The initial ideal of I_{2,(2,2),2} with x_1≻x_2 is (x_1^2, x_2^2, x_1x_2), so the non-leading term x_2^2 belongs to in(I_{n,m,k}), contradicting the lemma's assertion. The proof of Lemma 4.7 hinges on the claim that the truncation t_{\le n} of a tail monomial strictly divides s; this fails for tails of the form s'' x_n^N with N > s_n, precisely the situation in this counterexample. This is load-bearing: the proof of Theorem 4.8 invokes Lemma 4.7 directly to establish reducedness of the set in (9), and Proposition 4.9 and Theorem 4.11 also rely on it. The reduction clause in (10) appears to repair the particular example (the term x_2^2 is deleted), but Lemma 4.7 is not stated for the reduced polynomial. The lemma and its proof need to be rewritten for the post-reduction polynomials, or replaced by another argument, before the reducedness claim is established.
  2. [§4.2, Theorem 4.8] The proof of Theorem 4.8 asserts that "by Lemma 4.7, all its tail terms lie outside the initial ideal" for the polynomial g_s defined in (10). However, (10) includes an additional reduction modulo (x_j^{m_j},...,x_n^{m_n}), while Lemma 4.7 concerns the unreduced polynomial defined in Setup 4.2(i). The reduction step deletes monomials that lie in the initial ideal (as in the x_2^2 example), but the proof does not show that the reduced polynomial retains the property that all of its remaining tails are outside in(I_{n,m,k}), nor that the leading monomial is unchanged. Consequently, the conclusion that (9) is the reduced Gröbner basis does not follow from the stated lemmas. A revised proof must either prove the tail property directly for the reduced polynomials or show that the reduction operation preserves it.
minor comments (4)
  1. [§4.3, Example 4.12] In the displayed Gröbner basis element for s=x_1x_2x_3, the terms "1/2 x_1 x_2^4", "x_2 x_2^4", and "x_3 x_2^4" appear; since the ring has only four variables and the total degree is 4, these should presumably read "1/2 x_1 x_4^2", "x_2 x_4^2", and "x_3 x_4^2". As printed, the exponents are inconsistent and the terms are not homogeneous.
  2. [§4.1, proof of Theorem 4.5] There is an extra closing parenthesis in the expression "HS(R/(in(In,m,k))); t)"; it should be "HS(R/in(In,m,k); t)".
  3. [§3.2, Definition 3.18] The notation for the sets in Definition 3.18 and equation (4) is confusing: the definition introduces "Crit_{n,m,k,j}" with a prime, and then equation (4) uses "Crit_{n,m,k,j}" without a prime. The two symbols should be consistently distinguished, for instance by using "Crit'_{n,m,k,j}" throughout the definition and in equation (4).
  4. [§3.1, Remark 3.11] The formula d' = (Σ m_i)−n−d+k is written with unnecessary parentheses; the intended expression d' = Σ m_i − n − d + k would be clearer.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Gröbner basis derivation is self-contained; the self-citations appear only in auxiliary applications and do not feed the main theorem.

full rationale

The central result, Theorem 1.1, is derived from an explicit construction rather than from the desired conclusion. The initial ideal equality in(I_{n,m,k}) = (M_{n,m,k}) in Theorem 4.5 is proved by combining a Hilbert series comparison with explicit polynomials g_s whose leading term is the given monomial s. Proposition 3.14 computes the Hilbert series of R/(M_{n,m,k}) through the reflection map and the standard Gorenstein symmetry of monomial complete intersections; neither step imports the target Gröbner basis. Lemma 4.1 is a self-contained binomial identity, and the lower-bound comparison uses Fröberg's external theorem, not the paper's own claims. No parameter is fitted to data: all coefficients and exponents are explicit functions of the input m and k. The reflection map is defined from the socle data of R/P_{n,m}, not from the initial ideal of I_{n,m,k}, so the counting argument is genuinely independent. The paper does cite prior work by overlapping authors: [17] is used for the m=2 Catalan convolution and for counting Gröbner bases, and [22] supplies the characteristic-p WLP classification used in Section 5.5. These citations support applications and comparisons, not the proof of the main theorem, and they are external mathematical theorems rather than restatements of the present claim, so they do not make the derivation circular. The known proof gap in Lemma 4.7, where the asserted divisibility of truncations of tail monomials may fail for the raw g_s, is a correctness risk and would require a revised argument, but it is not circularity: it does not reduce the theorem to its own input. The paper's explicit acknowledgement that Proposition 5.13 is folklore with no located reference is likewise a citation gap, not a circular step, since a proof is supplied. Overall, no step in the chain rewrites the goal as an assumption or renames a fitted quantity as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation introduces no fitted free parameters. It relies on standard Gröbner-basis theory, Hilbert-series symmetry of Gorenstein monomial complete intersections, Fröberg's lower bound, and, in the application sections, on known classifications and a known log-concavity result. No new physical or algebraic entities are postulated; the red line and reflection map are proof devices defined from the input data.

assumptions (6)
  • standard math Fröberg lower bound: HS(R/In,m,k;t) is coefficientwise at least the truncated series [(1-t^k)HS(R/Pn,m;t)], and this series is the coefficientwise minimal Hilbert series for the quotient by n+1 forms of degrees (m1,...,mn,k).
    Used in the proof of Theorem 4.5 to identify in(In,m,k) with (Mn,m,k) by Hilbert-series comparison. The bound is attributed to Fröberg [11].
  • standard math The Hilbert series of R/Pn,m is symmetric and unimodal for Artinian monomial complete intersections.
    Used in Proposition 3.14 and Lemma 3.13 through the Gorenstein symmetry of the complete intersection; see Remark 2.2.
  • standard math Initial ideals preserve Hilbert series of quotient rings.
    Remark 2.1; used throughout to compare R/In,m,k with R/in(In,m,k).
  • standard math The socle-degree formula delta_{n-1} in Eq. (14) from Reid, Roberts and Roitman [26, Theorem 1] is correct.
    Used in Lemma 4.18, Proposition 4.20 and Corollary 4.22 in the enumerative and Lefschetz sections.
  • domain assumption For equigenerated complete intersections with n>=5, WLP in characteristic p is classified by p > floor((n(m-1)+1)/2), due to Kustin-Vraciu and Lundqvist-Nicklasson.
    Used in Theorem 5.14 to connect initial-ideal behaviour in characteristic p to WLP.
  • standard math The nontrivial part of the sequence of first differences of the Hilbert series of R/P_{2n,m} is log-concave, as established by Speyer [28].
    Used in Corollary 5.12 to derive log-concavity of rows of the (m-1)-Catalan triangle.

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Pith. "Pith review of The Gr\"obner basis for powers of a general linear form in a monomial complete intersection." pith.science (2026). https://pith.science/paper/E3UXSURV

@misc{pith2026250624028,
  author       = {Pith},
  title        = {Pith review of: The Gr\"obner basis for powers of a general linear form in a monomial complete intersection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3UXSURV}},
  note         = {Machine review of arXiv:2506.24028}
}
abstract

We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gr\"obner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gr\"obner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics.

Figures

Figures reproduced from arXiv: 2506.24028 by the authors.

Figure 1
Figure 1. Triangular array of Hilbert series coefficients together with the graph of the piecewise linear function Ln,m,k (shown in red), for parameters n = 4, m = (3, 2, 2, 3), and k = 2. When illustrating an m-admissible lattice path, we also display the graph of Ln,m,k, shown as a thin red piecewise linear curve. The following provides a first example. Example 3.6. Let m = (3, 2, 2, 3), and consider the m-free monomial t =… view at source ↗
Figure 2
Figure 2. The graph of L4,(3,2,2,3),2 and the lattice path associated with the monomial x1x3x 2 4 . We now introduce a reflection operation on lattice paths associated with the m-free mono￾mials. We will subsequently prove that those lattice paths whose reflections remain admissible correspond bijectively to the m-free monomials in the initial ideal of In,m,k. Definition 3.7. Let (a, b) ∈ Z 2 be a lattice point with 0 ≤ a ≤ n… view at source ↗
Figure 3
Figure 3. The lattice path P associated with the (4, 2, 5)-free monomial x 3 1x 2 3 and its reflection Λ(P). The reflected path corresponds to the monomial x 3 1x 4 3 . Definition 3.9. Let Mn,m,k be the set of monomials corresponding to critical m-admissible lattice paths. We write (Mn,m,k) for the ideal generated by the monomials in Mn,m,k + Pn,m. The reflection map Λ satisfies several properties, which we summarize in the f… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (Left) Schematic view of an edge AB between the vertical lines x = i − 1 and x = i in an m-admissible lattice path crossing the red line, starting above it and ending below. The slope of AB is at least 2 − mi . The positive integers δ and ϵ satisfy ϵ ≤ mi−1−2δ 2 . It f…
Figure 5
Figure 5. Figure 5: The lattice paths of the monomials in the minimal generating set S4,(3,2,2,3),2. Lemma 3.21. Let s ∈ Mn,m,k be a minimal generator of degree d. Let t be any monomial of degree d such that t ≻ s with respect to the reverse lexicographic ordering. Then t ∈ (Mn,m,k). Proo…
Figure 6
Figure 6. Figure 6: illustrates this transformation with an example. (0, 0) (0, −1) (4, 1) (0, 0) (0, 2) (1, 3) [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: The lattice path corresponding to the Gr¨obner basis leading term x1x2x 2 3x 2 5x6 ∈ in(I6,3,3), decomposed into translations of lattice paths associated with the leading terms x1x2x 2 3 ∈ in(I3,3,2) and x 2 2x3 ∈ in(I3,3,1). (0, 0) (5, −2) (0, 0) (2, −1) (0, 0) (3, −1…
Figure 8
Figure 8. Figure 8: The lattice path associated with the Gr¨obner basis leading term x1x 2 2x 2 4x 2 5 ∈ in(I5,3,4), decomposed into translations of lattice paths associated with the leading terms x1x 2 2 ∈ in(I2,3,2) and x 2 2x 2 3 ∈ in(I3,3,2). 5.3. Equigenerated case with k = 1 in quan…

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