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Hidefumi Ohsugi

Publications and source records attributed to Hidefumi Ohsugi.

At least 55 records · Page 3Linked to original sources
arXiv2013

Gorenstein cut polytopes

An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that Gorenstein cut polytopes are compressed (i.e., all of whose reverse lexicographic triangulations are unimodular). Second, by applying Athanasiadis's theory for Gorenstein compressed polytopes, we show that a cut polytope of a graph $G$ is Gorenstein if and only if $G$ has no $K_5$-minor and $G$ is either a bipartite graph without induced cycles of length $\geq 6$ or a bridgeless chordal graph.

math.CO↗
arXiv2013

Roots of the Ehrhart polynomial of hypersimplices

The Ehrhart polynomial of the $d$-th hypersimplex $Δ(d,n)$ 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 $Δ(d,n)$ 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 $Δ(d,n)$ satisfies $-\frac{n}{d} < {\rm Re} (a) < 1$ if $4 \leq d \ll n$.

math.CO↗
arXiv2013

Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses no quadratic Gröbner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for $I_{G}$ to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs $G$ with the above property, up to 8 vertices.

math.AC↗
arXiv2013

Markov chain Monte Carlo methods for the regular two-level fractional factorial designs and cut ideals

It is known that a Markov basis of the binary graph model of a graph $G$ corresponds to a set of binomial generators of cut ideals $I_{\widehat{G}}$ of the suspension $\widehat{G}$ of $G$. In this paper, we give another application of cut ideals to statistics. We show that a set of binomial generators of cut ideals is a Markov basis of some regular two-level fractional factorial design. As application, we give a Markov basis of degree 2 for designs defined by at most two relations.

math.ST↗
arXiv2012

Toric ideals and their circuits

In this paper, we study toric ideals generated by circuits. For toric ideals which have squarefree quadratic initial ideals, a sufficient condition to be generated by circuits is given. In particular, squarefree Veronese subrings, the second Veronese subrings and configurations arising from root systems satisfy the condition. In addition, we study toric ideals of finite graphs and characterize the graphs whose toric ideals are generated by circuits u -v such that either u or v is squarefree. There exists several classes of graphs whose toric ideals satisfy this condition and whose toric rings are nonnormal.

math.AC↗
arXiv2011

Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.

math.CO↗
arXiv2011

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, Gröbner 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.

math.AC↗
arXiv2010

Roots of Ehrhart polynomials arising from 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 {\it et al.}\ that all roots $α$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(α) \le D - 1$, 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 $|z+\tfrac{d}{4}| \le \tfrac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension $D$ has the roots of its Ehrhart polynomial in the narrower strip $-\tfrac{D}{2} \leq \Re(α) \leq \tfrac{D}{2}-1$. 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.

math.CO↗
arXiv2010

Toric rings and ideals of nested configurations

The toric ring together with the toric ideal arising from a nested configuration is studied, with particular attention given to the algebraic study of normality of the toric ring as well as the Gröbner bases of the toric ideal. One of the combinatorial applications of these algebraic findings leads to insights on smooth $3 \times 3$ transportation polytopes.

math.AC↗
arXiv2010

Roots of Ehrhart polynomials of Gorenstein Fano polytopes

Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.

math.CO↗
arXiv2009

Non-very ample configurations arising from contingency tables

In this paper, it is proved that, if a toric ideal possesses a fundamental binomial none of whose monomials is squarefree, then the corresponding semigroup ring is not very ample. Moreover, very ample semigroup rings of Lawrence type are discussed. As an application, we study very ampleness of configurations arising from contingency tables.

math.AC↗
arXiv2009

Normality of cut polytopes of graphs is a minor closed property

Sturmfels-Sullivant conjectured that the cut polytope of a graph is normal if and only if the graph has no K_5 minor. In the present paper, it is proved that the normality of cut polytopes of graphs is a minor closed property. By using this result, we have large classes of normal cut polytopes. Moreover, it turns out that, in order to study the conjecture, it is enough to consider 4-connected plane triangulations.

math.CO↗
arXiv2009

Powers of componentwise linear ideals

We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.

math.AC↗
arXiv2008

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.

math.AC↗
arXiv2008

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

math.AC↗
arXiv2008

Simple polytopes arising from finite graphs

Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Gröbner basis if the edge polytope ${\cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.

math.AC↗
arXiv2008

Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gröbner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.

math.ST↗