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Takayuki Hibi

Publications and source records attributed to Takayuki Hibi.

At least 19 recordsLinked to original sources
arXiv2026

Weakly Newton-nondegenerate binomial ideals

An ideal $I$ of a polynomial ring $R=k[x_1,\dots,x_n]$ is called weakly Newton-nondegenerate, or weakly NND, if its integral closure $\overline{I}$ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals \[ I=(x_1^2+\epsilon_1x_{a_1}x_{b_1},\ \dots,\ x_n^2+\epsilon_nx_{a_n}x_{b_n}),\qquad \epsilon_i\in\{\pm1\},\ a_i\neq b_i, \] over an algebraically closed field. We prove that $I$ is weakly NND if and only if $\overline I=\mathfrak{m}^2$, if and only if the given generators form a regular sequence, and if and only if an explicit combinatorial condition on the pair (support pattern, sign pattern) holds: no nonempty subset $S\subseteq\{1,\dots,n\}$ is simultaneously closed for the support data and sign-trivial for the associated lattice of relations. The last equivalence rests on a solvability criterion for systems of monomial equations over a divisible abelian group, in the spirit of Eisenbud and Sturmfels. As an application, we classify all such ideals for $n=3$.

math.AC↗
arXiv2026

Componentwise linear monomial ideals

Let $S=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with ${\rm deg} x_1=\cdots ={\rm deg} x_n = 1$ and ${\bf a} = (a_1,\ldots,a_n) \in {\mathbb Z}_{>0}^n$. Given a squarefree monomial $u=x_{i_1} \cdots x_{i_d}$ of $S$ with $1 \leq i_1 < \cdots < i_d \leq n$, we set $u^{[{\bf a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[{\bf a}]}$ with $G(I^{[{\bf a}]})=\{u^{[{\bf a}]} : u \in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[{\bf a}]}$ is studied.

math.AC↗
arXiv2026

Depth and Krull dimension of Binomial edge ideals

Let $J_G$ denote the binomial edge ideal of a finite graph $G$ in the polynomial ring $S$. We determine all triples $(n,t,d)$ with $n\geq 3$ for which there exists a finite connected graph $G$ on $n$ vertices with ${\mathrm{depth}}(S/J_{G})=t$ and $\dim(S/J_G)=d$.

math.AC↗
arXiv2026

Prime and Cohen--Macaulay binomial ideals with linear resolution

Let $I$ be an ideal of a polynomial ring $S$ over a field $K$ for which $I$ is minimally generated by at least two quadratic binomials. Suppose that (i) $I$ is prime, (ii) $S/I$ is Cohen--Macaulay and (iii) $I$ has linear resolution. The question whether $I$ is equal to the ideal of $2$-minors of a $(2 \times n)$-matrix of variables is mainly studied.

math.AC↗
arXiv2026

Depth of edge ideals and vertex connectivity of finite graphs

Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $κ(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq κ(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $κ(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $κ(G)$ is given.

math.AC↗
arXiv2026

Symmetric and unimodal independence polynomials of trees

Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.

math.CO↗
arXiv2026

Independence polynomials of graphs

In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we determine exactly when $P_G(-1)$ is $0, 1$, or $-1$, characterize the pseudo-Gorenstein$^*$ members, and show that there is a unique big star with symmetric independence polynomial. We also study graphs obtained from a graph $H$ by attaching leaves to selected vertices. We derive an explicit formula for the resulting independence polynomial, determine the corresponding value at $-1$, and prove that if every vertex of $H$ receives at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly two leaves. As an application, we obtain exact criteria for the values of $P_G(-1)$ and for the pseudo-Gorenstein$^*$ members of caterpillar graphs. For cochordal graphs, we classify all symmetric independence polynomials. Finally, for connected graphs on $n$ vertices with small independence numbers, we determine the exact range of possible values of $P_G(-1)$.

math.CO↗
arXiv2026

Join-meet binomial algebras of distributive lattices

We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated by quadrics is studied.

math.AC↗
arXiv2026

Scarf complexes of graphs and their powers

Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.

math.AC↗
arXiv2026

Cohen-Macaulay squares of edge ideals

Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic behaviour of powers of edge ideals. As an application, we demonstrate how Reisner's criterion can be applied directly to check if $I(G)^2$ is Cohen--Macaulay. We can show that if $G$ belongs to the class of finite graphs which consists of cycles, whisker graphs, trees, connected chordal graphs and connected Cohen--Macaulay bipartite graphs, then the square $I(G)^2$ is Cohen--Macaulay if and only if either $G$ is the pentagon, the cycle of length $5$, or $G$ consists of exactly one edge.

math.AC↗
arXiv2026

Pseudo-Gorenstein$^{*}$ Graphs

Motivated by pseudo-Gorenstein rings in commutative algebra, introduced by Herzog et al., we define pseudo-Gorenstein$^{*}$ graphs and classify them in several natural graph families using independence polynomials.

math.AC↗
arXiv2026

Partially ordered sets of distributive type and algebras with straightening laws

A finite poset (partially ordered set) $P$ with ${\hat 0}$ is called of distributive type if every interval $[{\hat 0}, a]$, $a \in P$, of $P$ is a distributive lattice. From a viewpoint of ASL's (algebras with straightening laws), the join-meet toric ring on a finite distributive lattice is generalized to an ASL on a finite poset of distributive type. Our target is the questions when a finite poset of distributive lattice is Cohen--Macaulay and when the ASL on it is Gorenstein. We focus on a natural class of finite posets of distributive type and study various aspects of the above questions.

math.AC↗
arXiv2026

Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

In this paper, we characterize all graphs $G$ satisfying \[\operatorname{reg}(S/J_G)=\ell(G)=c(G)\] where $\ell(G)$ is the sum of the lengths of the longest induced paths in each connected component of $G$ and $c(G)$ is the number of the maximal cliques of $G$. We also characterize all connected graphs $G$ that satisfy \[\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-ω(G)+1\] where $ω(G)$ is the clique number of $G$. Moreover, we investigate the possible values of the regularity of $S/J_G$ within the intervals $[\ell(G), c(G)]$ and $[\ell(G), |V(G)|-ω(G)+1]$.

math.AC↗
arXiv2026

Elimination ideals of Plücker ideals and algebras with straightening laws

It is well known that the Plücker ideal defining the Grassmannian is generated by quadratic Plücker relations. These relations form a reverse lexicographic Gröbner basis and endow the Plücker algebra with the structure of an algebra with straightening laws (ASL). In this paper, we study quadratically generated projections of the Grassmannian of lines $\mathrm{Gr}(2,n)$. We then combinatorially characterize the Gorenstein ASL subalgebras of the Plücker algebra of $\mathrm{Gr}(2,n)$.

math.AC↗
arXiv2025

Minimal primes and radicality of ideals generated by adjacent 2-minors

In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent $2$-minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.

math.AC↗
arXiv2025

Bounded powers of edge ideals: Pseudo-Gorenstein and Level polytopes

A lattice polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called level* if (i) $\mathcal{P}$ is normal, (ii) $(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n \neq \emptyset$ and (iii) for each $N = 2,3, \ldots$ and for each $\textbf{a} \in N(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$, there is $\textbf{a}_0 \in (\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$ together with $\textbf{a}' \in (N-1)\mathcal{P} \cap \mathbb{Z}^n$ for which $\textbf{a} = \textbf{a}_0 + \textbf{a}'$, where $N\mathcal{P} = \{N\textbf{a} : \textbf{a} \in \mathcal{P}\}$. A normal polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called pseudo-Gorenstein* [4] if $ |(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n| = 1. $ A pseudo-Gorenstein* polytope $\mathcal{P}$ is level* if and only if $\mathcal{P}$ is reflexive up to translation. In the present paper, level* polytopes together with pseudo-Gorenstein* polytopes arising from discrete polymatroids of bounded powers of edge ideals are studied.

math.AC↗