Related papers: Depth stability of cover ideals

We prove that the depth functions of cover ideals of balanced hyper- gragh have the non-increasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.

The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $p$ is the number of its…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,...,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. For every monomial ideal $I\subset S$, We provide a recursive formula to determine a lower bound for the…

Let $I$ be a square-free monomial ideal in $R = k[x_1,\ldots,x_n]$, and consider the sets of associated primes ${\rm Ass}(I^s)$ for all integers $s \geq 1$. Although it is known that the sets of associated primes of powers of $I$ eventually…

It is proved that if $I$ and $J$ are weakly stable ideals in a polynomial ring $R=k[x_1,\ldots,x_n]$, with $k$ a field, then the regularity of $\text{Tor}^R_i(R/I,R/J)$ has the expected upper bound. We also give a bound for the regularity…

We give upper bounds for the Stanley depth of edge ideals of certain k-partite clutters. In particular, we generalize a result of Ishaq about the Stanley depth of the edge ideal of a complete bipartite graph. A result of Pournaki, Seyed…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $I\subset S$ is a squarefree monomial ideal. For every integer $k\geq 1$, we denote the $k$-th symbolic…

Let $G=P_n$ be a path graph with cover ideal $J(P_n)$. By using Hochster's depth formula, we prove the explicit formulae to compute the depth functions of powers of cover ideals of paths.

Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S/I^t$ for $t\geq1$, where $I$ is the edge ideal of some caterpillar and…

Assume that $G$ is a graph with edge ideal $I(G)$. We provide sharp lower bounds for the depth of $I(G)^2$ in terms of the star packing number of $G$.

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals…

Let $S=K[x_1,...,x_n]$ be a polynomial ring in $n$ variables over the field $K$. For integers $1\leq t< n$ consider the ideal $I=(x_1,...,x_t)\cap(x_{t+1}, ...,x_n)$ in $S$. In this paper we bound from above the Stanley depth of the ideal…

Assume that $G$ is a graph with edge ideal $I(G)$ and star packing number $\alpha_2(G)$. We denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that the inequality ${\rm depth} S/(I(G)^{(s)})\geq \alpha_2(G)-s+1$ is true…

Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f \in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it does not have a root whose vector of the imaginary parts is contained in the interior…

We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals…

Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$…

Let $G$ be a finite simple graph and $J(G)$ denote its vertex cover ideal in a polynomial ring over a field. % $\mathbb{K}$. The $k$-th symbolic power of $J(G)$ is denoted by $J(G)^{(k)}$. In this paper, we give a criteria for cover ideals…

Edge ideals of finite simple graphs are well-studied over polynomial rings. In this paper, we initiate the study of edge ideals over exterior algebras, specifically focusing on the depth and singular varieties of such ideals. We prove an…

This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators…