Searcharxiv⌕ Search

arXiv subjects

Akiyoshi Tsuchiya

Publications and source records attributed to Akiyoshi Tsuchiya.

At least 37 records · Page 2Linked to original sources
arXiv2020

Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $A$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.

math.CO↗
arXiv2020

The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $γ$-positivity

A lattice polytope $\mathcal{P} \subset \mathbb{R}^d$ is called a locally anti-blocking polytope if for any closed orthant $\mathbb{R}^d_{\varepsilon}$ in $\mathbb{R}^d$, $\mathcal{P} \cap \mathbb{R}^d_{\varepsilon}$ is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. In the present paper, we give a formula for the $h^*$-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the $γ$-positivity of the $h^*$-polynomials of locally anti-blocking reflexive polytopes.

math.CO↗
arXiv2020

Reflexive polytopes arising from bipartite graphs with $γ$-positivity associated to interior polynomials

In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is reflexive if and only if $G$ is bipartite. Moreover, in the case, ${\mathcal B}_G$ has a regular unimodular triangulation. This implies that the $h^*$-polynomial of ${\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, namely the $γ$-positivity and the real-rootedness of the $h^*$-polynomials. In fact, if $G$ is bipartite, then the $h^*$-polynomial of ${\mathcal B}_G$ is $γ$-positive and its $γ$-polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $h^*$-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers--Stanley conjecture, we construct a bipartite graph $G$ whose $h^*$-polynomial is not real-rooted but $γ$-positive, and coincides with the $h$-polynomial of a flag triangulation of a sphere.

math.CO↗
arXiv2020

Enriched order polytopes and Enriched Hibi rings

Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope ${\mathcal O}_P$ and the chain polytope ${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ are equal to the order polynomial of $P$ that counts the $P$-partitions. In this paper, we introduce the enriched order polytope of a poset $P$ and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of $P$ and the left enriched order polynomial of $P$ that counts the left enriched $P$-partitions, by using the theory of Gröbner bases. The toric rings of enriched order polytopes are called enriched Hibi rings. It turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The above result implies the existence of a bijection between the lattice points in the dilations of ${\mathcal O}^{(e)}_P$ and ${\mathcal C}^{(e)}_P$. Towards such a bijection, we give the facet representations of enriched order and chain polytopes.

math.CO↗
arXiv2019

Odd cycles and Hilbert functions of their toric rings

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that, for each $1 \leq s \leq 5$, an $O$-sequence $(h_0, h_1, \ldots, h_{2s-1}) \in \mathbb{Z}_{\geq 0}^{2s}$ satisfying the properties that (i) $h_0 \leq h_1 \leq \cdots \leq h_{s-1}$, (ii) $h_{2s-1} = h_0$, $h_{2s-2} = h_1$ and (iii) $h_{2s - 1 - i} = h_i + (-1)^{i}$, $2 \leq i \leq s - 1$, can be the $h$-vector of a Cohen--Macaulay standard $G$-domain.

math.AC↗
arXiv2019

Nef-partitions arising from unimodular configurations

Reflexive polytopes have been studied from viewpoints of combinatorics, commutative algebra and algebraic geometry. A nef-partition of a reflexive polytope $\mathcal{P}$ is a decomposition $\mathcal{P}=\mathcal{P}_1+\cdots+\mathcal{P}_r$ such that each $\mathcal{P}_i$ is a lattice polytope containing the origin. Batyrev and van Straten gave a combinatorial method for explicit constructions of mirror pairs of Calabi-Yau complete intersections obtained from nef-partitions. In the present paper, by means of Gröbner basis techniques, we give a large family of nef-partitions arising from unimodular configurations.

math.CO↗
arXiv2019

Edge rings of bipartite graphs with linear resolutions

Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a $2$-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a $q$-linear resolution, where $q \geq 3$, is a hypersurface and proved the case $q=3$. In the present paper, we solve this conjecture for the case of finite connected simple bipartite graphs.

math.AC↗
arXiv2019

Depth of an initial ideal

Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0 \leq r \leq d$, there exists a monomial order $<_r$ on $S$ with ${\rm depth} (S/{\rm in}_{<_r}(I)) = r$, where ${\rm in}_{<_r}(I)$ is the initial ideal of $I$ with respect to $<_r$.

math.AC↗
arXiv2019

Gorenstein simplices with a given $δ$-polynomial

To classify the lattice polytopes with a given $δ$-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their $δ$-polynomials are of the form $1+t^k+\cdots+t^{(v-1)k}$, where $k$ and $v$ are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above $δ$-polynomials will be performed, when $v$ is either $p^2$ or $pq$, where $p$ and $q$ are prime integers with $p \neq q$. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected $δ$-polynomial.

math.CO↗
arXiv2019

Enriched chain polytopes

Stanley introduced a lattice polytope $\mathcal{C}_P$ arising from a finite poset $P$, which is called the chain polytope of $P$. The geometric structure of $\mathcal{C}_P$ has good relations with the combinatorial structure of $P$. In particular, the Ehrhart polynomial of $\mathcal{C}_P$ is given by the order polynomial of $P$. In the present paper, associated to $P$, we introduce a lattice polytope $\mathcal{E}_{P}$, which is called the enriched chain polytope of $P$, and investigate geometric and combinatorial properties of this polytope. By virtue of the algebraic technique on Gröbner bases, we see that $\mathcal{E}_P$ is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the $h^*$-polynomial of $\mathcal{E}_P$ is equal to the $h$-polynomial of a flag triangulation of a sphere. On the other hand, by showing that the Ehrhart polynomial of $\mathcal{E}_P$ coincides with the left enriched order polynomial of $P$, it follows from works of Stembridge and Petersen that the $h^*$-polynomial of $\mathcal{E}_P$ is $γ$-positive. Stronger, we prove that the $γ$-polynomial of $\mathcal{E}_P$ is equal to the $f$-polynomial of a flag simplicial complex.

math.CO↗
arXiv2019

Classification of lattice polytopes with small volumes

In the frame of a classification of general square systems of polynomial equations solvable by radicals, Esterov and Gusev succeeded in classifying all spanning lattice polytopes whose normalized volumes are at most $4$. In the present paper, we complete to classify all lattice polytopes whose normalized volumes are at most $4$ based on the known classification of their $δ$-polynomials.

math.CO↗
arXiv2019

The depth of a reflexive polytope

Given arbitrary integers $d$ and $r$ with $d \geq 4$ and $1 \leq r \leq d + 1$, a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ with ${\rm depth} K[\mathcal{P}] = r$ for which its dual polytope $\mathcal{P}^\vee$ is normal will be constructed, where $K[\mathcal{P}]$ is the toric ring of $\mathcal{P}$.

math.AC↗
arXiv2019

Integer decomposition property for Cayley sums of order and stable set polytopes

Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytope and the stable set polytope of a graph, the following conditions are equivalent: (i) the Cayley sum is Gorenstein; (ii) the Minkowski sum is Gorenstein; (iii) the graph is perfect.

math.CO↗
arXiv2019

Regularity and $a$-invariant of Cameron--Walker graphs

Let $S$ be the polynomial ring over a field $K$ and $I \subset S$ a homogeneous ideal. Let $h(S/I,λ)$ be the $h$-polynomial of $S/I$ and $s = \mathrm{deg} h(S/I,λ)$ the degree of $h(S/I,λ)$. It follows that the inequality $s - r \leq d - e$, where $r = \mathrm{reg} (S/I)$, $d = \dim S/I$ and $e = \mathrm{depth} S/I$, is satisfied and, in addition, the equality $s - r = d - e$ holds if and only if $S/I$ has a unique extremal Betti number. We are interested in finding a natural class of finite simple graphs $G$ for which $S/I(G)$, where $I(G)$ is the edge ideal of $G$, satisfies $s - r = d - e$. Let $a(S/I(G))$ denote the $a$-invariant of $S/I$, i.e., $a(S/I(G)) = s - d$. One has $a(S/I(G)) \leq 0$. In the present paper, by showing the fundamental fact that every Cameron--Walker graph $G$ satisfies $a(S/I(G)) = 0$, a class of Cameron--Walker graphs $G$ for which $S/I(G)$ satisfies $s - r = d - e$ will be exhibited.

math.AC↗
arXiv2018

Reflexive polytopes arising from edge polytopes

It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every $(0,1)$-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large family of $(0,1)$-polytopes are the edge polytopes of finite simple graphs. In the present paper, it is shown that, by giving a new class of reflexive polytopes, each edge polytope is unimodularly equivalent to a facet of some reflexive polytope. Furthermore, we extend the characterization of normal edge polytopes to a characterization of normality for these new reflexive polytopes.

math.CO↗
arXiv2018

Stanley's non-Ehrhart-positive order polytopes

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart polynomial are positive. Answering an Ehrhart positivity question posed on Mathoverflow, Stanley provided an example of a non-Ehrhart-positive order polytope of dimension $21$. Stanley's example comes from a certain family of order polytopes. In this paper, we study the Ehrhart positivity question on this family of polytopes. By giving explicit formulas for the coefficients of the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we determine the sign of each Ehrhart coefficient of each polytope in the family. As a consequence of our result, we conclude that for any positive integer $d \ge 21,$ there exists an order polytope of dimension $d$ that is not Ehrhart positive, and for any positive integer $\ell$, there exists an order polytope whose Ehrhart polynomial has precisely $\ell$ negative coefficients, which answers a question posed by Hibi. We finish this article by discussing the existence of lower-dimensional order polytopes whose Ehrhart polynomials have a negative coefficient.

math.CO↗