Takayuki Hibi
Depth of initial ideals of normal edge rings.
Mon, 01/24/2011 - 16:01 — SomNambulAuthors: Takayuki Hibi, Akihiro Higashitani, Kyouko Kimura, Augustine B. O'Keefe
Subjects: Commutative Algebra
AbstractLet $G$ be a finite graph on the vertex set $[d] = \{1, ..., d \}$ with the
edges $e_1, ..., e_n$ and $K[\tb] = K[t_1, ..., t_d]$ 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 $\tb^e = t_it_j$ such that $e =
\{i, j\}$ is an edge of $G$. Let $K[\xb] = K[x_1, ..., x_n]$ be the polynomial
ring in $n$ variables over $K$ and define the surjective homomorphism $\pi :
K[\xb] \to K[G]$ by setting $\pi(x_i) = \tb^{e_i}$ for $i = 1, ..., n$. The
toric ideal $I_G$ of $G$ is the kernel of $\pi$.- Read more
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Depth of edge rings arising from finite graphs.
Thu, 09/09/2010 - 15:01 — SomNambulAuthors: Takayuki Hibi, Akihiro Higashitani, Kyouko Kimura, Augustine B. O'Keefe
Subjects: Commutative Algebra
AbstractLet $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the
technique of Gr\"obner bases and initial ideals, it will be proved that, given
integers $f$ and $d$ with $7 \leq f \leq d$, there exists a finite graph $G$ on
$[d]={1,...,d}$ with $\depth K[G] = f$ and with $\Krull-dim K[G] = d$.Roots of Ehrhart polynomials of Gorenstein Fano polytopes.
Tue, 01/26/2010 - 16:01 — SomNambulAuthors: Takayuki Hibi, Akihiro Higashitani, Hidefumi Ohsugi
Subjects: Combinatorics
AbstractGiven 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 $(-1, 0)$.Non-very ample configurations arising from contingency tables.
Mon, 11/23/2009 - 16:01 — SomNambulAuthors: Takayuki Hibi, Hidefumi Ohsugi
Subjects: Commutative Algebra
AbstractIn 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.Binomial edge ideals.
Mon, 09/28/2009 - 15:24 — SomNambulAuthors: Takayuki Hibi, Juergen Herzog, Freyja Hreinsdottir
Subjects: Commutative Algebra
AbstractWe introduce binomial edge ideals attached to a simple graph and study their
algebraic properties.Smooth Fano polytopes arising from finite partially ordered sets.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Takayuki Hibi, Akihiro Higashitani
Subjects: Combinatorics
AbstractGorenstein Fano polytopes arising from finite partially ordered sets will be
introduced. Then we study the problem which partially ordered sets yield smooth
Fano polytopes.- 2 comments
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