Sums of powers of linear forms, and gorenstein algebras

1999 ◽  
pp. 57-72 ◽  
Author(s):  
Anthony Iarrobino ◽  
Vassil Kanev
Keyword(s):  
Linear Forms ◽  
Gorenstein Algebras ◽  
Sums Of Powers ◽  
Powers Of Linear Forms

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

Lattice Octahedra and Sums of Powers of Linear Forms

1988 ◽  
Vol s2-38 (2) ◽  
pp. 207-215 ◽  
Author(s):  
T. W. Cusick ◽  
J. Wolfskill
Keyword(s):  
Linear Forms ◽  
Sums Of Powers ◽  
Powers Of Linear Forms

Decomposition of quantics in sums of powers of linear forms

1996 ◽  
Vol 53 (2-3) ◽  
pp. 93-107 ◽  
Author(s):  
P. Comon ◽  
B. Mourrain
Keyword(s):  
Linear Forms ◽  
Sums Of Powers ◽  
Powers Of Linear Forms

On Sums of Powers of Linear Forms. II

1949 ◽  
Vol 50 (3) ◽  
pp. 699 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Linear Forms ◽  
Sums Of Powers ◽  
Powers Of Linear Forms

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.

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.)

scholarly journals Identifiability of homogeneous polynomials and Cremona transformations

2019 ◽  
Vol 2019 (757) ◽  
pp. 279-308 ◽  
Author(s):  
Francesco Galuppi ◽  
Massimiliano Mella
Keyword(s):  
Special Class ◽  
Homogeneous Polynomial ◽  
Projective Spaces ◽  
Homogeneous Polynomials ◽  
Additive Decomposition ◽  
Linear Forms ◽  
Cremona Transformations ◽  
General Polynomial ◽  
Powers Of Linear Forms

AbstractA homogeneous polynomial of degree d in {n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in {n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

scholarly journals Linear Dependence Of Powers Of Linear Forms

2015 ◽  
Vol 29 (1) ◽  
pp. 131-138
Author(s):  
Andrzej Sładek
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Dependence ◽  
Linear Forms ◽  
Powers Of Linear Forms

AbstractThe main goal of the paper is to examine the dimension of the vector space spanned by powers of linear forms. We also find a lower bound for the number of summands in the presentation of zero form as a sum of d-th powers of linear forms.

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