A lower bound for Stanley depth of squarefree monomial ideals

In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It follows easily that the Stanley Conjecture holds for the factor if and only if it holds for its canonical form. In particular, we construct an algorithm which simplifies the depth computation and using the canonical form we massively reduce the run time for the sdepth computation.

Mathematics

Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish general closed formulas for Stanley depth of quotient of edge ideals associated with the m t h -power of a wheel graph, for m ≥ 3 , gear graphs and anti-web gear graphs.

We give some precise formulas for the depth of the quotient rings of the edge ideals associated to a graph consisting, either of the union of some line graphs L 3r 1 ,. .. , L 3r k 1 , L 3s 1 +1 ,. .. , L 3s k 2 +1 and L 3t 1 +2 ,. .. , L 3t k 3 +2 or of the union of cycle graphs C 3r 1 ,. .. , C 3r k 1 , C 3s 1 +1 ,. .. , C 3s k 2 +1 and C 3t 1 +2 ,. .. , C 3t k 3 +2 , with a common vertex. We also give some tight bounds for their Stanley depths. Mathematics Subject Classification (2010): 13C15; 13P10; 13F20.

arXiv (Cornell University), 2023

Let I n,m := (x 1 x 2 • • • x m , x 2 x 3 • • • x m+1 ,. .. , x n−m+1 • • • x n) be the m-path ideal of the path graph of length n, in the ring S = K[x 1 ,. .. , x n ]. We prove that: depth(S/I t n,m) = n − t + 2 − n−t+2 m+1 − n−t+2 m+1 , t ≤ n + 1 − m m − 1, t > n + 1 − m , for all t ≥ 1. Also, we prove that depth(S/I n,m) ≥ sdepth(S/I t n,m) ≥ depth(S/I t n,m) and sdepth(I t n,m) ≥ depth(I t n,m), for all t ≥ 1.

Communications in Algebra, 2012

Let I ⊂ J be monomial ideals of a polynomial algebra S over a field. Then the Stanley depth of J/I is smaller or equal with the Stanley depth of √ J/ √ I. We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q ′ , which is reached if Q, Q ′ are irreducible, ht(Q + Q ′ ) is odd and √ Q, √ Q ′ have no common variable.

2013

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt cons...

Hacettepe Journal of Mathematics and Statistics

In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for values of depth and Stanley depth for some subclasses. We also give some sharp upper bounds for depth and Stanley depth in the general cases.

Commutative Algebra, 2012

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.

2012

Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's conjecture holds for a square free monomial ideal then it holds for all its trivial modifications.