arXiv subjects

Akihiro Higashitani

Publications and source records attributed to Akihiro Higashitani.

At least 91 records · Page 5Linked to original sources

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↗

arXiv2016

Almost Gorenstein homogeneous rings and their $h$-vectors

In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the $h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.

math.AC↗

arXiv2015

Dominating induced matchings of finite graphs and regularity of edge ideals

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which forms a maximal matching, then the induced matching number of $G$ is equal to the minimum matching number of $G$. In the present paper, from viewpoints of both combinatorics and commutative algebra, finite simple graphs with dominating induced matchings will be mainly studied.

math.CO↗

arXiv2015

Unimodality on $δ$-vectors of lattice polytopes and two related properties

In this paper, we investigate two properties concerning the unimodality of the $δ$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove that the dilated lattice polytopes $n\mathcal{P}$ have strictly log-concave and strictly alternatingly increasing $δ$-vectors if $n > \max\{s,d+1-s\}$, where $s$ is the degree of the $δ$-polynomial of $\mathcal{P}$. The bound $\max\{s,d+1-s\}$ for $n$ is reasonable. We also provide several kinds of unimodal (or non-unimodal) $δ$-vectors. Concretely, we give examples of lattice polytoeps whose $δ$-vectors are not unimodal, unimodal but neither log-concave nor alternatingly increasing, alternatingly increasing but not log-concave, and log-concave but not alternatingly increasing, respectively.

math.CO↗

arXiv2015

Equivalence classes for smooth Fano polytopes

Let $\mathcal{F}(n)$ be the set of smooth Fano $n$-polytopes up to unimocular equivalence. In this paper, we consider the F-equivalence or I-equivalence classes for $\mathcal{F}(n)$ and introduce F-isolated or I-isolated smooth Fano $n$-polytopes. First, we describe all of F-equivalence classes and I-equivalence classes for $\mathcal{F}(5)$. We also give a complete characterization of F-equivalence classes (I-equivalence classes) for smooth Fano $n$-polytopes with $n+3$ vertices and construct a family of I-isolated smooth Fano polytopes.

math.AG↗

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↗

arXiv2015

Ehrhart polynomial roots of reflexive polytopes

Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.

math.CO↗

arXiv2014

Integer decomposition property of free sums of convex polytopes

Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.

math.CO↗

arXiv2014

Algebraic study on Cameron-Walker graphs

Let $G$ be a finite simple graph on $[n]$ and $I(G) \subset S$ the edge ideal of $G$, where $S = K[x_{1}, \ldots, x_{n}]$ is the polynomial ring over a field $K$. Let $m(G)$ denote the maximum size of matchings of $G$ and $im(G)$ that of induced matchings of $G$. It is known that $im(G) \leq \text{reg}(S/I(G)) \leq m(G)$, where $\text{reg}(S/I(G))$ is the Castelnuovo-Mumford regularity of $S/I(G)$. Cameron and Walker succeeded in classifying the finite connected simple graphs $G$ with $im(G) = m(G)$. We say that a finite connected simple graph $G$ is a Cameron-Walker graph if $im(G) = m(G)$ and if $G$ is neither a star nor a star triangle. In the present paper, we study Cameron-Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron-Walker graph $G$ is unmixed if and only if $G$ is Cohen-Macaulay and classify all Cohen-Macaulay Cameron-Walker graphs. Second, we prove that there is no Gorenstein Cameron-Walker graph. Finally, we prove that every Cameron--Walker graph is sequentially Cohen-Macaulay.

math.AC↗

arXiv2014

Minkowski sum of polytopes and its normality

In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.

math.CO↗

arXiv2014

Smooth Fano polytopes arising from finite directed graphs

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit Kähler--Einstein metrics.

math.CO↗

arXiv2013

Ehrhart polynomials with negative coefficients

math.CO↗

arXiv2013

Normal cyclic polytopes and cyclic polytopes that are not very ample

Let $d$ and $n$ be positive integers with $n \geq d + 1$ and $τ_{1}, ..., τ_{n}$ integers with $τ_{1} < ... < τ_{n}$. Let $C_{d}(τ_{1}, ..., τ_{n}) \subset \RR^{d}$ denote the cyclic polytope of dimension $d$ with $n$ vertices $(τ_{1},τ_{1}^{2},...,τ_{1}^{d}), ..., (τ_{n},τ_{n}^{2},...,τ_{n}^{d})$. We are interested in finding the smallest integer $γ_{d}$ such that if $τ_{i+1} - τ_{i} \geq γ_{d}$ for $1 \leq i < n$, then $C_{d}(τ_{1}, ..., τ_{n})$ is normal. One of the known results is $γ_{d} \leq d (d + 1)$. In the present paper a new inequality $γ_{d} \leq d^{2} - 1$ is proved. Moreover, it is shown that if $d \geq 4$ with $τ_{3} - τ_{2} = 1$, then $C_{d}(τ_{1}, ..., τ_{n})$ is not very ample.

math.CO↗

arXiv2013

Integer decomposition property of dilated polytopes

Let $\mathcal{P} \subset \mathbb{R}^N$ be an integral convex polytope of dimension $d$ and write $k \mathcal{P}$, where $k = 1, 2, \ldots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k = 1, 2, \ldots$ and for any $α\in k \mathcal{P} \cap \mathbb{Z}^N$, there exist $α_{1}, \ldots, α_k$ belonging to $\mathcal{P} \cap \mathbb{Z}^N$ such that $α= α_1 + \cdots + α_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

math.CO↗

arXiv2012

Non-normal very ample polytopes and their holes

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

math.CO↗

arXiv2012

Roots of Ehrhart polynomials and symmetric $δ$-vectors

The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. \cite[Conjecture 4.10]{MHNOH}, says that all roots $α$ of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension $d$ satisfy $-\frac{d}{2} \leq \Re(α) \leq \frac{d}{2} -1$. In this paper, we observe the behaviors of roots of SSNN polynomials which are a wider class of the polynomials containing all the Ehrhart polynomials of Gorenstein Fano polytopes. As a result, we verify that this conjecture is true when the roots are real numbers or when $d \leq 5$.

math.CO↗

arXiv2012

Ehrhart polynomials of integral simplices with prime volumes

For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call $δ(\Pc)=(δ_0, δ_1,..., δ_d)$ the $δ$-vector of $\Pc$ and $\vol(\Pc)=\sum_{i=0}^dδ_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $δ$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $δ$-vectors of integral simplices with normalized volume 5 and 7.

math.CO↗

arXiv2012

Toric rings arising from cyclic polytopes

In the present paper, we consider the problem when the toric ring arising from an integral cyclic polytope is Cohen-Macaulay by discussing Serre's condition and we give a complete characterization when that is Gorenstein. Moreover, we study the normality of the other semigroup ring arising from an integral cyclic polytope but generated only with its vertices.

math.AC↗