Related papers: A note on Stanley conjecture for monomial ideals

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture…

In this paper, we partially confirm a conjecture, proposed by Cimpoea\c{s}, Keller, Shen, Streib and Young, on the Stanley depth of squarefree Veronese ideals $I_{n,d}$. This conjecture suggests that, for positive integers $1 \le d \le n$,…

Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ holds. In the…

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 8$ or $n\leq 10$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)-1$.

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be monomial ideal of $R$. In this paper, we show that if $I$ is a generic monomial ideal, then $R/I$ is pretty clean if and only if $R/I$ is…

Given a nonincreasing function $f : \mathbb{Z}_{\geq 0} \setminus \{ 0 \} \to \mathbb{Z}_{\geq 0}$ such that (i) $f(k) - f(k+1) \leq 1$ for all $k \geq 1$ and (ii) if $a = f(1)$ and $b = \lim_{k \to \infty} f(k)$, then $|f^{-1}(a)| \leq…

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for…

Let $I$ be a monomial ideal of $S=K[x_1,\ldots,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\operatorname{hdepth}(I)=n$, (iii) $\operatorname{hdepth}(S/I)=n-1$. Assuming that $I$ is squarefree, we prove that…

We have introduced the Janet's algorithm for the Stanley decomposition of a monomial ideal I in a polynomial ring S = K[x_1,...,x_n] and prove that Janet's algorithm gives the squarefree Stanley decomposition of S/I for a squarefree…

Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let ${\rm sgn}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R/I$ is the depth of $R/{\rm sgn}(I)$, and…

We consider the path ideal associated to a line graph, we compute \texttt{sdepth} for its quotient ring and note that it is equal with its \texttt{depth}. In particular, it satisfies the Stanley inequality.

We extend a result of Minh and Trung to get criteria for $\depth I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial complexes…

Let $\varphi : S = k[y_0,..., y_n] \to R = k[y_0,...,y_n]$ be given by $y_i \to f_i$ where $f_0,...,f_n$ is an $R$-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, $I_\Delta \subseteq S$, of…

We give tight bounds for the Stanley depth of the quotient ring of the path ideal of a cycle graph. In particular, we prove that it satisfies the Stanley inequality.

Under suitable technical assumptions, a description is given for the generators of $s$-residual intersections of an ideal $I$ in terms of lower residual intersections, if $s \geq \mu(I)-2$. This implies that $s$-residual intersections can…

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is…

We give sharp bounds for the Stanley depth of a special class of ideals of Borel type.

We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.

The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$…

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds…