均匀拟阵的Speyer $g$-多项式序列的无限对数凹性与高阶Turán不等式
Infinite log-concavity and higher order Turán inequality for the sequences of Speyer's $g$-polynomial of uniform matroids
AI总结：
本文研究均匀拟阵的Speyer $g$-多项式序列，通过证明其生成函数在$t>0$时仅含实零点，得到该序列的无限对数凹性、高阶Turán不等式及多项相关性质，丰富了Turán型不等式领域的研究。
摘要：
设$U_{n,d}$为$n$个元素上秩为$d$的均匀拟阵，记$g_{U_{n,d}}(t)$为$U_{n,d}$的Speyer $g$-多项式。Turán不等式与高阶Turán不等式和实整函数的Laguerre-Pólya（$\u2112$-$\u2131$）类相关，而$\u2112$-$\u2131$类与黎曼猜想联系紧密，Turán型不等式已受到广泛关注，无限对数凹性则是Turán不等式在另一方向上的深刻推广。本文主要得到：对任意$t>0$，序列$\u007bg_{U_{n,d}}(t)\u007d_{d=1}^{n-1}$满足无限对数凹性与高阶Turán不等式。为证明这些结果，我们证明了$g_{U_{n,d}}(t)$的生成函数（记为$h_n(x;t)$）在$t>0$时仅含实零点。由此还得到：$t>0$时$h_n(x;t)$的$\u03b3$-正性、$g_{U_{n,d}}(t)$的渐近正态性，以及$g_{U_{n,d}}(t)$与$h_n(x;t)$的Laguerre不等式。
英文摘要：
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 Turán inequality and higher order Turán inequality are related to the Laguerre-Pólya ($\mathcal{L}$-$\mathcal{P}$) class of real entire functions, and the $\mathcal{L}$-$\mathcal{P}$ class has close relation with the Riemann hypothesis. The Turán type inequalities have received much attention. Infinite log-concavity is also a deep generalization of Turán inequality with different direction. In this paper, we mainly obtain the infinite log-concavity and the higher order Turán inequality of the sequence $\{g_{U_{n,d}}(t)\}_{d=1}^{n-1}$ for any $t>0$. 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 $γ$-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 $h_n(x;t)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
