English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basisgenerating functions for all matroids. From the viewpoint of combinatorialHodge theory, it is natural to ask whether these functions are "strictly"log-concave for simple matroids. In this paper, we show this strictness forsimple graphic matroids, that is, we show that Kirchhoff polynomials of simplegraphs are strictly log-concave. Our key observation is that the Kirchhoffpolynomial {of a complete graph} can be seen as the (irreducible) relativeinvariant of a certain prehomogeneous vector space, which may be independentlyinteresting in its own right. Furthermore, we prove that for any$a_i\in\mathbb{R}_{>0}$, $a_1x_1+\cdots+a_nx_n\in R^1_{M}$ satisfies the strongLefschetz property (moreover, Hodge--Riemann bilinear relation) at degree oneof the Artinian Gorenstein algebra $R^*_M$ associated to a graphic matroid $M$,which is defined by Maeno and Numata for all matroids.

Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property