scholarly journals Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Upper Bound ◽  
Linear Span ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
Secant Varieties ◽  
Secant Variety ◽  
Symmetric Tensor Rank ◽  
Tangent Vectors ◽  
Minimal Integer

Let X⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈 〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg.

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 Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 140
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear Span ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
General Point ◽  
Symmetric Tensor Rank ◽  
Maximum Rank ◽  
Veronese Embedding ◽  
Finite Set

Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | ≤ ( n + ⌊ d / 2 ⌋ n ) . For the tensor rank, we describe the cases with | S | ≤ 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d).

scholarly journals Seeking for the Maximum Symmetric Rank

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 247 ◽  
Author(s):  
Alessandro De Paris
Keyword(s):  
State Of The Art ◽  
Symmetric Tensor ◽  
Upper Bounds ◽  
The State ◽  
Tensor Rank ◽  
Symmetric Tensors ◽  
Lower And Upper Bounds ◽  
Symmetric Tensor Rank ◽  
Set Up ◽  
Divided Powers

We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.

scholarly journals Symmetric Tensors and Symmetric Tensor Rank

2008 ◽  
Vol 30 (3) ◽  
pp. 1254-1279 ◽  
Author(s):  
Pierre Comon ◽  
Gene Golub ◽  
Lek-Heng Lim ◽  
Bernard Mourrain
Keyword(s):  
Symmetric Tensor ◽  
Tensor Rank ◽  
Symmetric Tensors ◽  
Symmetric Tensor Rank
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