Related papers: On stability properties of powers of polymatroidal…

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this…

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $K$ and $I$ be a matroidal ideal of degree $d$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integers $l$ and $k$, for which $\Ass(I^l)$ and $\depth(R/I^k)$…

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest…

In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in…

Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\depth S/I_G^k$ stabilizes is denoted by $\dstab(I_G)$. We show that $\dstab(I_G)<\ell(I_G)$ where $\ell(I_G)$ denotes the…

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest…

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…

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 $I$ be a monomial ideal of a polynomial ring $R$. In this paper we determine a number $B$ such that $\Ass (I^n/I^{n+1}) = \Ass (I^{B}/I^{B+1})$ for all $n\geq B$.

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. In this paper, it is shown that Stanley's conjecture holds for $S/I$, if $I$ is a weakly polymatroidal ideal.

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured…

The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal…

Let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $k$ and $I$ be a matroidal ideal of degree $d$. In this paper, we study the unmixedness properties and the arithmetical rank of $I$. Moreover, we show that…

The sequence $(\operatorname{Ass}(R/I^n))_{n\in\mathbb{N}}$ of associated primes of powers of a monomial ideal $I$ in a polynomial ring $R$ eventually stabilizes by a known result by Markus Brodmann. L\^e Tu\^an Hoa gives an upper bound for…

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…

Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize…

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 R = K[x1,...,xr] be a polynomial ring over a field K. Let G be a graph with vertex set {1,...,r} and let J be the cover ideal of G. We give a sharp bound for the stability index of symbolic depth function sdstab(J). In the case G is…

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…

The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring consist of the functions that vanish up to order $n$ in the variety defined by $I$. These do not necessarily coincide with the ordinary algebraic powers $I^n$, but it…