Centrally symmetric configurations of integer matrices

The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal as well as is Gorenstein. In addition, Gr\"obner bases of toric ideals of centrally symmetric configurations will be discussed. Special attentions will be given to centrally symmetric configurations of unimodular matrices and those of incidence matrices of finite graphs.Comment: 14 pages, 1 figur

Two way subtable sum problems and quadratic Groebner bases

Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable sum problems and shows that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Groebner bases.Comment: 3 page

Roots of the Ehrhart polynomial of hypersimplices

The Ehrhart polynomial of the $d$-th hypersimplex $\Delta(d,n)$ of order $n$ is studied. By computational experiments and a known result for $d=2$, we conjecture that the real part of every roots of the Ehrhart polynomial of $\Delta(d,n)$ is negative and larger than $- \frac{n}{d}$ if $n \geq 2d$. In this paper, we show that the conjecture is true when $d=3$ and that every root $a$ of the Ehrhart polynomial of $\Delta(d,n)$ satisfies $-\frac{n}{d} < {\rm Re} (a) < 1$ if $4 \leq d \ll n$.Comment: 18 pages, 8 figure