Adam Schweitzer | Searcharxiv

arXiv2026

A note on the ultra log-concavity of matroid intersection

In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Br\"and\'en and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In this note, we make the next step by proving it for $M_2^\natural$-concave functions. This shows the same property for the intersection of any pair of matroids (which itself may not be a matroid). Furthermore, we show that this can not be further extended to the intersections of three matroids by including a counterexample of partition matroids.

math.CO↗

arXiv2026

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

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.

math.CO↗