Akihiro Higashitani

Counterexamples of the Conjecture on Roots of Ehrhart Polynomials

On roots of Ehrhart polynomials, Beck et al. conjecture that all roots α of the Ehrhart polynomial of an integral convex polytope of dimension d satisfy −d≤ℜ(α)≤d−1. In this paper, we provide counterexamples for this conjecture.

Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e 1,…, e n and K[t] = K[t 1,…, t d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t e = t i t j such that e = {i, j} is an edge of G. Let K[x] = K[x 1,…, x n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x i ) = t e i for i = 1,…, n. The toric ideal I G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <rev on K[x] and a lexicographic order <lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in<rev (I G ) = f and K[x]/in<lex (I G ) is Cohen–Macaulay, where in<rev (I G ) (resp., in<lex (I G )) is the initial ideal of I G with respect to <rev (resp., <lex) and where depth K[x]/in<rev (I G ) is the depth of K[x]/in<rev (I G ).

Lattice multi-polygons

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice

NORMAL CYCLIC POLYTOPES AND CYCLIC POLYTOPES THAT ARE NOT VERY AMPLE

\mathbb{Z}^2

Roots of Ehrhart polynomials of Gorenstein Fano polytopes

. We first prove a formula on the rotation number of a unimodular sequence in

Smooth Fano Polytopes Arising from Finite Partially Ordered Sets

\mathbb{Z}^2

Unimodality of \(\delta \)-vectors of lattice polytopes and two related properties

. This formula implies the generalized twelve-point theorem in [12]. We then introduce the notion of lattice multi-polygons which is a generalization of lattice polygons, state the generalized Picks formula and discuss the classification of Ehrhart polynomials of lattice multi-polygons and also of several natural subfamilies of lattice multi-polygons.

Integer Decomposition Property of Free Sums of Convex Polytopes

Let

Toric Rings Arising from Cyclic Polytopes

d

Existence of regular unimodular triangulations of dilated empty simplices

and