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Koutarou Yoshida

Publications and source records attributed to Koutarou Yoshida.

4 recordsLinked to original sources
arXiv2019

Gorenstein simplices with a given $δ$-polynomial

To classify the lattice polytopes with a given $δ$-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their $δ$-polynomials are of the form $1+t^k+\cdots+t^{(v-1)k}$, where $k$ and $v$ are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above $δ$-polynomials will be performed, when $v$ is either $p^2$ or $pq$, where $p$ and $q$ are prime integers with $p \neq q$. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected $δ$-polynomial.

math.CO↗
arXiv2017

Existence of regular unimodular triangulations of dilated empty simplices

Given integers $k$ and $m$ with $k \geq 2$ and $m \geq 2$, let $P$ be an empty simplex of dimension $(2k-1)$ whose $δ$-polynomial is of the form $1+(m-1)t^k$. In the present paper, the necessary and sufficient condition for the $k$-th dilation $kP$ of $P$ to have a regular unimodular triangulation will be presented.

math.CO↗
arXiv2016

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. Moreover, it is also shown that for each $d \geq 3$ and $1 \leq k \leq d-2$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that the coefficient of $n^k$ of the Ehrhart polynomial $i(\mathcal{P},n)$ of $\mathcal{P}$ is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.

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↗