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arXiv · 2601.00782

Existence of K\"ahler algebras with Chow polynomials as Hilbert series

Adam Schweitzer·Lorenzo Vecchi
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Abstract

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the K\"ahler package such that their Hilbert--Poincar\'e series agrees with the Chow polynomial. Our statement provides evidence in support of a conjecture by Ferroni, Matherne and the second author about the existence of an algebra for every weakly ranked poset that generalizes the Feichtner--Yuzvinsky Chow ring for matroids. This allows us to prove strong inequalities for the coefficients of Chow polynomials; we prove log-concavity for all posets of weak rank at most six and provide counterexamples to log-concavity for any higher rank. For ranked posets we recover an even stronger condition, showing that the differences between consecutive coefficients constitute a pure O-sequence.

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Adam Schweitzer, Lorenzo Vecchi. 2026-01-02. Existence of K\"ahler algebras with Chow polynomials as Hilbert series. https://arxiv.org/abs/2601.00782

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