Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Stanley depth of edge ideals of k-partite clutters
66
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 Fakhari and Yassemi implies that the Stanleys conjecture holds for d-uniform complete d-partite clutters. Here we give a shorter and different proof of this fact.
rate research
Read More
Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $sdepth(M)$, and conjectured that $depth(M) le sdepth(M)$ for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1le dle n < 5d+4$, then $sdepth(I_{n,d})= floor{binom{n}{d+1}Big/binom{n}{d}}+d$, and if $dgeq 1$ and $nge 5d+4$, then $d+3le sdepth(I_{n,d}) le floor{binom{n}{d+1}Big/binom{n}{d}}+d$.
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 analytic spread of $I$. For trees we give a stronger upper bound for $dstab(I_G)$. We also show for any two integers $1leq a<b$ there exists a tree for which $dstab(I_G)=a$ and $ell(I_G)=b$.
Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornuejols and to max-flow min-cut problems.
Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen-Macaulay defect of $S/J_G$ in terms of Cohen-Macaulay defect of $S_H/J_H$ and using this we construct a graph with Cohen-Macaulay defect $q$ for any $qgeq 1$. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair $(r,b)$ of positive integers with $1leq b< r$, there exists a connected graph $G$ such that $reg(S/J_G)=r$ and the number of extremal Betti number of $S/J_G$ is $b$.
In this article we associate to every lattice ideal $I_{L,rho}subset K[x_1,..., x_m]$ a cone $sigma $ and a graph $G_{sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $sigma $. To every polynomial $F$ we assign a subgraph $G_{sigma}(F)$ of the graph $G_{sigma}$. Every expression of the radical of $I_{L,rho}$, as a radical of an ideal generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of $G_{sigma}$, the $cup_{i=1}^s G_{sigma}(F_i)$. This result provides a lower bound for the minimal number of generators of $I_{L,rho}$ and therefore improves the generalized Krulls principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the $A$-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a family of examples, that the bounds given are sharp.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
