Strictness of the log-concavity of generating polynomials of matroids

Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequaliti...

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Bibliographic Details
Published inJournal of combinatorial theory. Series A Vol. 181; p. 105351
Main Authors Murai, Satoshi, Nagaoka, Takahiro, Yazawa, Akiko
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.07.2021
Subjects
Hodge–Riemann relation
Independent set
Lorentzian polynomial
Mason's conjecture
Matroid
Morphism of matroids
Lorentzian polynomial
Independent set
Mason's conjecture
Morphism of matroids
Matroid
Hodge–Riemann relation
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Summary:Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2020.105351