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Akiyoshi Tsuchiya

Publications and source records attributed to Akiyoshi Tsuchiya.

57 records · Page 4Linked to original sources

arXiv2015

Best possible lower bounds on the coefficients of Ehrhart polynomials

For an integral convex polytope $\mathcal{P} \subset \mathbb{R}^d$, we recall $L_\mathcal{P}(n)=|n\mathcal{P} \cap \mathbb{Z}^d|$ the Ehrhart polynomial of $\mathcal{P}$. Let $g_r(\mathcal{P})$ be the $r$th coefficients of $L_\mathcal{P}(n)$ for $r=0,\ldots,d$. Martin Henk and Makoto Tagami gave lower bounds on the coefficients $g_r(\mathcal{P})$ in terms of the volume of $\mathcal{P}$. They proved that these bounds are best possible for $r \in \{1,2,d-2\}$. We show that these bounds are also optimal for $r=3$ and $d-r$ even and we give a new best possible bound for $r=d-3$.

math.CO↗

arXiv2015

Gorenstein polytopes with trinomial $h^*$-polynomials

The characterization of lattice polytopes based upon information about their Ehrhart $h^*$-polynomials is a difficult open problem. In this paper, we finish the classification of lattice polytopes whose $h^*$-polynomials satisfy two properties: they are palindromic (so the polytope is Gorenstein) and they consist of precisely three terms. This extends the classification of Gorenstein polytopes of degree two due to Batyrev and Juny. The proof relies on the recent characterization of Batyrev and Hofscheier of empty lattice simplices whose $h^*$-polynomials have precisely two terms. Putting our theorem in perspective, we give a summary of these and other existing results in this area.

math.CO↗

arXiv2013

Ehrhart polynomials with negative coefficients

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative.

math.CO↗