Year: 2025
Type: article
Abstract: A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated with bimatroids. Bimatroids can be used to characterize morphisms of matroids and this observation (originally due to Kung) allows us to prove a weak version of logarithmic concavity of the number of bases of a morphism of matroids. This is weaker than the original result by Eur and Huh; it nevertheless provides us with a new perspective on Mason’s log-concavity conjecture for independent sets of matroids. We finally show that for realizable bimatroids, the regular minor polynomial is a volume polynomial. Applied to morphisms of matroids, this shows that the weak basis generating polynomial of a morphism is a volume polynomial; this confirms a conjecture of Eur–Huh for morphisms of nullity $$\le 1$$ ≤ 1 and gives an algebro-geometric explanation for Mason’s log-concavity conjecture in the realizable case.
Source: Annals of Combinatorics
Authors: Felix Röhrle, Martin Ulirsch
Institutions: University of Tübingen, Paderborn University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: Deutsche Forschungsgemeinschaft
Awards: G251152387, Von Riemannschen Flächen zu tropischen Kurven (und wieder zurück), G5846448348, Tropische Lineare Algebra neu gedacht: Gebäude, Bimatroide, und Anwendungen, G6601682123, TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie, TRR 326: Geometrie und Arithmetik uniformisierter Strukturen (GAUS)
OpenAlex ID: W4413382161