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Takayuki Hibi

Publications and source records attributed to Takayuki Hibi.

At least 181 records · Page 10Linked to original sources
arXiv2008

The depth of an ideal with a given Hilbert function

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given a numerical function $H : \mathbb{N} \to \mathbb{N}$ satisfying $H=H_{A/I}$ for some homogeneous ideal $I$ of $A$, we write $\mathcal{A}_H$ for the set of those integers $0 \leq r \leq n$ such that there exists a homogeneous ideal $I$ of $A$ with $H_{A/I} = H$ and with $\depth A / I = r$. It will be proved that one has either $\mathcal{A}_H = \{0, 1, ..., b \}$ for some $0 \leq b \leq n$ or $|\mathcal{A}_H| = 1$.

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

Algebraic shifting and graded Betti numbers

Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$. Let $Δ$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_Δ\subset S$ its Stanley--Reisner ideal. We write $Δ^e$ for the exterior algebraic shifted complex of $Δ$ and $Δ^c$ for a combinatorial shifted complex of $Δ$. Let $β_{ii+j}(I_Δ) = \dim_K \Tor_i(K, I_Δ)_{i+j}$ denote the graded Betti numbers of $I_Δ$. In the present paper it will be proved that (i) $β_{ii+j}(I_{Δ^e}) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $β_{ii+j}(I_Δ) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is arbitrary. Thus in particular one has $β_{ii+j}(I_Δ) \leq β_{ii+j}(I_{Δ^{lex}})$ for all $i$ and $j$, where $Δ^{lex}$ is the unique lexsegment simplicial complex with the same $f$-vector as $Δ$ and where the base field is arbitrary.

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↗
arXiv2008

Groebner bases of nested configurations

In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.

math.AC↗
arXiv2007

Gotzmann ideals of the polynomial ring

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$. We will classify all the Gotzmann ideals of $A$ with at most $n$ generators. In addition, we will study Hilbert functions $H$ for which all homogeneous ideals of $A$ with the Hilbert function $H$ have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notion of segmentwise critical Hilbert function and show that segmentwise critical Hilbert functions are inflexible.

math.AC↗
arXiv2007

Linear balls and the multiplicity conjecture

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented.

math.AC↗
arXiv2007

Vertex cover algebras of unimodular hypergraphs

It is proved that all vertex cover algebras of a hypergraph are standard graded if and only if the hypergraph is unimodular. This has interesting consequences on the symbolic powers of monomial ideals.

math.AC↗
arXiv2006

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

math.AC↗
arXiv2006

Standard graded vertex cover algebras, cycles and leaves

The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.

math.AC↗
arXiv2006

Gin and Lex of certain monomial ideals

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic 0 with each $°x_i = 1$. Given arbitrary integers $i$ and $j$ with $2 \leq i \leq n$ and $3 \leq j \leq n$, we will construct a monomial ideal $I \subset A$ such that (i) $β_k(I) < β_k(\Gin(I))$ for all $k < i$, (ii) $β_i(I) = β_i(\Gin(I))$, (iii) $β_\ell(\Gin(I)) < β_\ell(\Lex(I))$ for all $\ell < j$ and (iv) $β_j(\Gin(I)) = β_j(\Lex(I))$, where $\Gin(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1 > >... > x_n$ and where $\Lex(I)$ is the lexsegment ideal with the same Hilbert function as $I$.

math.AC↗
arXiv2005

Special simplices and Gorenstein toric rings

Christos Athanasiadis studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal $h$-vectors arising from finite graphs.

math.AC↗
arXiv2004

The depth of powers of an ideal

We study the limit and initial behavior of the numerical function $f(k)=\depth S/I^k$. General properties of this function together with concrete examples arising from combinatorics are discussed.

math.AC↗
arXiv2004

Cohen-Macaulay Polymatroidal Ideals

All Cohen--Macaulay polymatroidal ideals are classified. The Cohen--Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the squarefree Veronese ideals.

math.AC↗