TY - JOUR
T1 - Strictness of the log-concavity of generating polynomials of matroids
AU - Murai, Satoshi
AU - Nagaoka, Takahiro
AU - Yazawa, Akiko
N1 - Funding Information:
The authors wish to express their gratitude to Yasuhide Numata for fruitful discussions. The authors also thank the referee for helpful comments and suggestions. The research of the first author is partially supported by KAKENHI 16K05102 , and the research of the second author is partially supported by Grant–in–Aid for JSPS Fellows 19J11207 .
Publisher Copyright:
© 2020
PY - 2021/7
Y1 - 2021/7
N2 - 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.
AB - 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.
KW - Hodge–Riemann relation
KW - Independent set
KW - Lorentzian polynomial
KW - Mason's conjecture
KW - Matroid
KW - Morphism of matroids
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U2 - 10.1016/j.jcta.2020.105351
DO - 10.1016/j.jcta.2020.105351
M3 - Article
AN - SCOPUS:85101208525
SN - 0097-3165
VL - 181
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
M1 - 105351
ER -