scholarly journals Monomial ideals and the failure of the Strong Lefschetz property

2021 ◽  
Author(s):  
Nasrin Altafi ◽  
Samuel Lundqvist
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Hilbert Functions ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Sharp Lower Bound

AbstractWe give a sharp lower bound for the Hilbert function in degree d of artinian quotients $$\Bbbk [x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree $$d \ge 2$$ d ≥ 2 . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

scholarly journals Stanley’s conjecture for critical ideals

2011 ◽  
Vol 48 (2) ◽  
pp. 220-226
Author(s):  
Azeem Haider ◽  
Sardar Khan
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth

Let S = K[x1,…,xn] be a polynomial ring in n variables over a field K. Stanley’s conjecture holds for the modules I and S/I, when I ⊂ S is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non-critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.

Sharp lower bounds on the sum-connectivity index of quasi-tree graphs

2021 ◽  
pp. 2150136
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Tree Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. A graph [Formula: see text] is called quasi-tree, if there exists [Formula: see text] such that [Formula: see text] is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs.

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

ON THE STANLEY DEPTH AND SIZE OF MONOMIAL IDEALS

2017 ◽  
Vol 59 (3) ◽  
pp. 705-715
Author(s):  
S. A. SEYED FAKHARI
Keyword(s):  
Lower Bound ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Recursive Formula ◽  
Monomial Ideals ◽  
Stanley Depth

AbstractLet $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

scholarly journals Classification of rational surfaces of degree 11 and sectional genus 11 in ${\mathsf P}^{4}$

2009 ◽  
Vol 104 (1) ◽  
pp. 60 ◽  
Author(s):  
Hans-Christian Graf von Bothmer ◽  
Kristian Ranestad
Keyword(s):  
Hilbert Function ◽  
Hilbert Functions ◽  
Sectional Genus ◽  
Rational Surfaces ◽  
The Third

We use the BGG-correspondence to show that there are at most three possible Hilbert functions for smooth rational surfaces of degree 11 and sectional genus 11. Surfaces with one of these Hilbert functions have been classified by Popescu. The classification for a second one is done in this paper. For the third Hilbert function the classification is still open.

scholarly journals Sharp Lower Bounds of the Sum-Connectivity Index of Unicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Maryam Atapour
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Unicyclic Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in graph G , respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D G ≥ 5 .

Lower Bounds for Matrices, II

1991 ◽  
Vol 44 (1) ◽  
pp. 54-74 ◽  
Author(s):  
Grahame Bennett
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Monotonicity Property ◽  
Hausdorff Matrix ◽  
Increasing Function ◽  
Sharp Lower Bound

AbstractOur main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: thenis an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

scholarly journals Gin and lex of certain monomial ideals

2006 ◽  
Vol 99 (1) ◽  
pp. 76 ◽  
Author(s):  
Satoshi Murai ◽  
Takayuki Hibi
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Lexicographic Order ◽  
Monomial Ideals ◽  
Initial Ideal ◽  
Generic Initial Ideal ◽  
Lexsegment Ideal

Let $A = K[x_1,\ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic $0$ with each $\deg x_i = 1$. Given arbitrary integers $i$ and $j$ with $2 \leq i \leq n$ and $3 \leq j \leq n$, we will construct a monomial ideal $I \subset A$ such that (i) $\beta_k(I) < \beta_k(\mathrm{Gin}(I))$ for all $k < i$, (ii) $\beta_i(I)= \beta_i(\mathrm{Gin}(I))$, (iii) $\beta_\ell((\mathrm{Gin}(I)) < \beta_\ell((\mathrm{Lex}(I))$ for all $\ell < j$ and (iv) $\beta_j(\mathrm{Gin}(I)) = \beta_j(\mathrm{Lex}(I))$, where $\mathrm{Gin}(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1 > \cdots > x_n$ and where $\mathrm{Lex}(I)$ is the lexsegment ideal with the same Hilbert function as $I$.

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