Representations of homogeneous polynomials as a sum of powers of linear forms with restricted support

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.

Download Full-text

Real and Complex Rank for Real Symmetric Tensors with Low Ranks

Algebra ◽

10.1155/2013/794054 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 2

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.

Download Full-text

A general canonical expression

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100011476 ◽

1933 ◽

Vol 29 (4) ◽

pp. 465-469 ◽

Cited By ~ 4

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..

Download Full-text

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

Geometry ◽

10.1155/2013/715907 ◽

2013 ◽

Vol 2013 ◽

pp. 1-2 ◽

Cited By ~ 3

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.)

Download Full-text

On Sums of Powers of Linear Forms. I

Annals of Mathematics ◽

10.2307/1969556 ◽

1949 ◽

Vol 50 (3) ◽

pp. 691 ◽

Cited By ~ 9

Author(s):

R. A. Rankin

Keyword(s):

Linear Forms ◽

Sums Of Powers ◽

Powers Of Linear Forms

Download Full-text

Lattice Octahedra and Sums of Powers of Linear Forms

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-38.2.207 ◽

1988 ◽

Vol s2-38 (2) ◽

pp. 207-215 ◽

Cited By ~ 3

Author(s):

T. W. Cusick ◽

J. Wolfskill

Keyword(s):

Linear Forms ◽

Sums Of Powers ◽

Powers Of Linear Forms

Download Full-text

Linear Dependence Of Powers Of Linear Forms

Annales Mathematicae Silesianae ◽

10.1515/amsil-2015-0010 ◽

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.

Download Full-text

Hierarchical Zonotopal Power Ideals

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2939 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Matthias Lenz

Keyword(s):

Characteristic Polynomial ◽

Hilbert Series ◽

Finite Sequence ◽

Vector Spaces ◽

Geometric Structures ◽

Linear Forms ◽

International Audience ◽

Powers Of Linear Forms ◽

Matroid Structure

International audience Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu. La théorie de l'algèbre "zonotopique'' s'occupe d'idéaux et d'espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite $X$, un nombre entier $k \geq -1$ et un ensemble supérieur dans le treillis des plans du matroïde défini par $X$, nous définissons et étudions l'$\textit{idéal hiérarchique zonotopique}$, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de $X$. Il existe des relations avec d'autres invariants de matroïdes, tels que le polynôme d'épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d'Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l'algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.

Download Full-text