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Bounds on the Sperner number and the flatness of the h-vector for binomial Macaulay dual generators

Determine quantitative bounds for the Sperner number S_{A_F} and for NS_{A_F}, the number of degrees attaining S_{A_F}, of the Artinian Gorenstein algebra A_F over a field of characteristic zero with binomial Macaulay dual generator F = X_1^{a_1}⋯X_n^{a_n}(X_1^{b_1}⋯X_r^{b_r} − X_{r+1}^{b_{r+1}}⋯X_n^{b_n}) and 1 ≤ r ≤ n − 1.

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Background

The Sperner number and the length of the flat portion of the h-vector play a central role in Lefschetz properties and their transfer to quotients. This paper establishes flats for certain families and proves transfer results (e.g., when F is obtained by differentiating another generator), but general bounds in terms of binomial exponents are not yet known.

The authors identify this as an open problem necessary to understand how WLP/SLP can be ensured or propagated in broader settings.

References

In this last section we would like to formulate the open problems appearing in the introduction in the concrete case of AG algebras having binomial Macaulay dual generator. In the authors solved all the above problems in the codimension 3 case ($n=3$), while for arbitrary codimenson the problems are largely open, although partial results to some of them are given in previous sections of this paper.

— New families of Artinian Gorenstein algebras with the weak Lefschetz property (2502.16687 - Altafi et al., 23 Feb 2025) in Section 4, Open problems