Convex bodies: the brunn–minkowski theory

Computing the Continuous Discretely

Cohen‐macaulay modules over cohen‐macaulay rings

The topology of toric varieties

In this paper we study product graphs and obtain some results. Wealso indicate the scope of its applications in a variety of fields.

On product graphs

Algebraic study on Cameron–Walker graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots ? of Ehrhart polynomials of polytopes of dimension D satisfy ?D?Re(?)?D?1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle $|z+\frac{d}{4}| \le \frac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip $-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1$ . Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.

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Roots of Ehrhart polynomials arising from graphs

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Grobner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the -vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

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Arithmetic aspects of symmetric edge polytopes

An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ such that $\alpha = \alpha_{1} + \cdots + \alpha_k$. 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.

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Integer decomposition property of dilated polytopes

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

Akihiro Higashitani

Papers

Algebraic study on Cameron–Walker graphs

Roots of Ehrhart polynomials arising from graphs

Arithmetic aspects of symmetric edge polytopes

Integer decomposition property of dilated polytopes

Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions.