Searcharxiv⌕ Search

arXiv subjects

Akihiro Higashitani

Publications and source records attributed to Akihiro Higashitani.

99 records · Page 6Linked to original sources
arXiv2011

Hermite normal forms and $δ$-vector

Let $δ(\Pc) = (δ_0, δ_1,..., δ_d)$ be the $δ$-vector of an integral polytope $\Pc \subset \RR^N$ of dimension $d$. Following the previous work of characterizing the $δ$-vectors with $\sum_{i=0}^d δ_i \leq 3$, the possible $δ$-vectors with $\sum_{i=0}^d δ_i = 4$ will be classified. And each possible $δ$-vectors can be obtained by simplices. We get this result by studying the problem of classifying the possible integral simplices with a given $δ$-vector $(δ_0, δ_1,..., δ_d)$, where $\sum_{i=0}^d δ_i \leq 4$, by means of Hermite normal forms of square matrices.

math.CO↗
arXiv2011

Counterexamples of the conjecture on roots of Ehrhart polynomials

An outstanding conjecture on roots of Ehrhart polynomials says that all roots $α$ of the Ehrhart polynomial of an integral convex polytope of dimension $d$ satisfy $-d \leq \Re(α) \leq d-1$. In this paper, we suggest some counterexamples of this conjecture.

math.CO↗
arXiv2011

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 $K[\tb] = K[t_1, ..., t_d]$ 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 $π: K[\xb] \to K[G]$ by setting $π(x_i) = \tb^{e_i}$ 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 $6 \leq f \leq d$, there exist a finite connected nonbipartite graph $G$ on $[d]$ together with a reverse lexicographic order $<_{\rev}$ on $K[\xb]$ and a lexicographic order $<_{\lex}$ on $K[\xb]$ such that (i) $K[G]$ is normal, (ii) $\depth K[\xb]/\ini_{<_{\rev}}(I_G) = f$ and (iii) $K[\xb]/\ini_{<_{\lex}}(I_G)$ is Cohen--Macaulay, where $\ini_{<_{\rev}}(I_G)$ (resp.\ $\ini_{<_{\lex}}(I_G)$) is the initial ideal of $I_G$ with respect to $<_{\rev}$ (resp.\ $<_{\lex}$) and where $\depth K[\xb]/\ini_{<_{\rev}}(I_G)$ is the depth of $K[\xb]/\ini_{<_{\rev}}(I_G)$.

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

Depth of edge rings arising from finite graphs

Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gröbner 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 $\Krull-dim K[G] = d$.

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↗
arXiv2010

Shifted symmetric $δ$-vectors of convex polytopes

A $δ$-vector $δ(\Pc)= (δ_0, δ_1, ..., δ_d)$ is called shifted symmetric if $δ_{d-i} = δ_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. A natural family of $(0,1)$-polytopes with shifted symmetric $δ$-vectors will be studied.

math.CO↗