Infinite log-concavity and higher order Turán inequality for the sequences of Speyer's $g$-polynomial of uniform matroids
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arXiv
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2024
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| _version_ | 1866909343961579520 |
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| author | Zhao, James J. Y. |
| author_facet | Zhao, James J. Y. |
| contents | 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 $h_n(x;t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_08085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite log-concavity and higher order Turán inequality for the sequences of Speyer's $g$-polynomial of uniform matroids Zhao, James J. Y. Combinatorics 05B35, 26C10 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 $h_n(x;t)$. |
| title | Infinite log-concavity and higher order Turán inequality for the sequences of Speyer's $g$-polynomial of uniform matroids |
| topic | Combinatorics 05B35, 26C10 |
| url | https://arxiv.org/abs/2409.08085 |