On the Ranks and Border Ranks of Symmetric Tensors

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by consideri...

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Vydané v:	Foundations of computational mathematics Ročník 10; číslo 3; s. 339 - 366
Hlavní autori:	Landsberg, J. M., Teitler, Zach
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer-Verlag 01.06.2010 Springer Nature B.V
Predmet:	Applications of Mathematics Computer Science Economics Geometry Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Polynomials Border rank 15A69 14N15 Secant varieties Symmetric tensor rank 15A21
ISSN:	1615-3375, 1615-3383
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Popis
Shrnutí:	Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.
Bibliografia:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14
ISSN:	1615-3375 1615-3383
DOI:	10.1007/s10208-009-9055-3