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Minimal free resolutions for binomial Macaulay dual generators

Determine minimal free resolutions, including graded Betti numbers and module structures, for the Artinian Gorenstein algebra A_F over a field of characteristic zero whose Macaulay dual generator is the binomial 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}) with 1 ≤ r ≤ n − 1.

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Background

The paper gives structural insights and connected-sum decompositions for some binomial families, but the homological picture—explicit minimal free resolutions—remains largely unresolved. For monomial complete intersections, Koszul complexes provide resolutions, but binomial cases need dedicated analysis.

Identifying resolutions would clarify homological properties and could leverage decompositions via doubling and connected sums as suggested by the authors.

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