bibkey: miroroigtran2020weak authors: Rosa M. Miró-Roig and Quang Hoa Tran year: 2020 title: On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms doi: 10.1016/j.jalgebra.2019.12.029 url: https://arxiv.org/abs/2001.06143v1 claim: The conjecture immediately after Proposition 3.12 states that the displayed alternating binomial coefficient is strictly negative for every integer n at least two. strata_touched:

• D5/S3/Combinatorics/Splines/MiroRoigTranStrictSign license: citation-only triage: anchor

Miró-Roig and Tran on uniform powers of general linear forms

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Rosa M. Miró-Roig and Quang Hoa Tran, On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms, Journal of Algebra 551 (2020), 209-231, DOI 10.1016/j.jalgebra.2019.12.029, primary source https://arxiv.org/abs/2001.06143v1.

The claim and its scope were read in that arXiv v1 body. The journal citation and DOI identify the final Elsevier version of record as metadata; its body was not read.

Immediately after the proof of Proposition 3.12, the authors conjecture that for every integer n>=2,

sum_{k=0}^n (-1)^k binom(2n+2,k)
  * (2n^2 - 1 - (2n-1)k)^(2n-1) < 0.

The n>=4 hypothesis in Proposition 3.12(c) belongs to the preceding weak-Lefschetz implication. It does not narrow the quantifier on this separate conjectural display. The authors report computations for 2<=n<=400 and say that the values suggest a monotonicity pattern; neither the finite computation, the monotonicity suggestion, nor the WLP consequence is the assertion cited by the formal result.

Formal use

D5/S3/Combinatorics/Splines/MiroRoigTranStrictSign.coefficient transcribes the display as an integer sum. result proves its strict negativity for every natural n>=2. The analytic spline recurrence and curvature modules are repo-derived proof infrastructure and are not attributed to this article.

Bounded later-source status

The refreshed bounded inspection covered Boij and Lundqvist, arXiv 2010.01107v2, Section 4; Booth, Singh and Vraciu, arXiv 2410.22542v3; and Boij and Lundqvist, arXiv 2608.22823v1, including Lemma 3.3, Theorem 3.4 and Remark 3.5. Those sources address broader almost-complete-intersection classifications, fixed-degree weak Lefschetz questions, Hilbert-series bounds, Cremona transformations, and WLP counterexamples. No proof of this exact all-n strict coefficient inequality was found in that inspected scope. This is a bounded source finding, not a claim of worldwide absence, priority, or exhaustive citation coverage.