scholarly journals On the weak Lefschetz property for powers of linear forms

2012 ◽  
Vol 6 (3) ◽  
pp. 487-526 ◽  
Author(s):  
Juan C. Migliore ◽  
Rosa M. Miró-Roig ◽  
Uwe Nagel
Keyword(s):  
Linear Forms ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Powers Of Linear Forms

scholarly journals Lefschetz property and powers of linear forms in 𝕂[x,y,z]

2018 ◽  
Vol 30 (4) ◽  
pp. 857-865 ◽  
Author(s):  
Charles Almeida ◽  
Aline V. Andrade
Keyword(s):  
Linear Form ◽  
Characteristic Zero ◽  
Maximal Rank ◽  
General Linear ◽  
Linear Forms ◽  
Lefschetz Property ◽  
Powers Of Linear Forms

Abstract In [9], Migliore, Miró-Roig and Nagel proved that if {R=\mathbb{K}[x,y,z]} , where {\mathbb{K}} is a field of characteristic zero, and {I=(L_{1}^{a_{1}},\dots,L_{4}^{a_{4}})} is an ideal generated by powers of four general linear forms, then the multiplication by the square {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of {R/I} . More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if {I=(L_{1}^{a_{1}},\dots,L_{r}^{a_{r}})} is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of {R/I} .

scholarly journals On the weak Lefschetz property for vector bundles on P2

2021 ◽  
Vol 568 ◽  
pp. 22-34
Author(s):  
Gioia Failla ◽  
Zachary Flores ◽  
Chris Peterson
Keyword(s):  
Vector Bundles ◽  
Weak Lefschetz Property ◽  
Lefschetz Property

scholarly journals Real and Complex Rank for Real Symmetric Tensors with Low Ranks

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Edoardo Ballico ◽  
Alessandra Bernardi
Keyword(s):  
Homogeneous Polynomial ◽  
Symmetric Tensors ◽  
Linear Forms ◽  
The Real ◽  
The Difference ◽  
Powers Of Linear Forms

We study the case of a real homogeneous polynomial whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

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.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

scholarly journals An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-2 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Upper Bound ◽  
Homogeneous Polynomial ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
Degree Polynomial ◽  
Linear Forms ◽  
Symmetric Tensor Rank ◽  
Low Degree ◽  
Powers Of Linear Forms

Fix integers m≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.)

On Sums of Powers of Linear Forms. I

1949 ◽  
Vol 50 (3) ◽  
pp. 691 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Linear Forms ◽  
Sums Of Powers ◽  
Powers Of Linear Forms
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