Infinite log-concavity and higher order Turán inequality for Speyer's $g$-polynomial of uniform matroids

Abstract

Let $U_{n,d}$ be the uniform matroid of rank $d$ on $n$ elements. Denote by $g_{U_{n,d}}(t)$ the Speyer's $g$-polynomial of $U_{n,d}$. The Tur\'{a}n inequality and higher order Tur\'{a}n inequality are related to the Laguerre-P\'{o}lya ($\mathcal{L}$-$\mathcal{P}$) class of real entire functions, and the $\mathcal{L}$-$\mathcal{P}$ class has close relation with the Riemann hypothesis. The Tur\'{a}n type inequalities have received much attention. Infinite log-concavity is also a deep generalization of Tur\'{a}n inequality with different direction. In this paper, we mainly obtain the infinite log-concavity and the higher order Tur\'{a}n inequality of the sequence $\{g_{U_{n,d}}(t)\}_{d=1}^{n-1}$ for $t>0$. In order to prove these results, we show that the generating function of $g_{U_{n,d}}(t)$, denoted $h_n(x;t)$, has only real zeros for $t>0$. Consequently, for $t>0$, we also obtain the $\gamma$-positivity of the polynomial $h_n(x;t)$, the asymptotical normality of $g_{U_{n,d}}(t)$, and the Laguerre inequalities for $g_{U_{n,d}}(t)$ and $h_n(x;t)$.