Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support. (29th December 2013)

• Record Type: Journal Article
• Authors: Ballico, E.
• Other Names: Cegarra A. M. Academic Editor.; Farnsteiner R. Academic Editor.; Gallego F. B. Academic Editor.; Kar S. K. Academic Editor.; Schenzel P. Academic Editor.
• Is Part Of: ISRN geometry. Volume 2013(2013)
• Publication Date: 2013-12-29
• Subjects: Geometry -- Periodicals; Topology -- Periodicals; Geometry; Topology; Periodicals; 516
• DOI: 10.1155/2013/704072
• Languages: English
• ISSNs: 2090-6307

Abstract

Abstract : Let ν d : ℙ m → ℙ n, n : = ( n + d n ) - 1, denote the degree d Veronese embedding of ℙ m . For any P ∈ ℙ n, let s r ( P ) be the minimal cardinality of S ⊂ ν d ( ℙ m ) such that P ∈ 〈 S 〉 . Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯ ( S ) powers L d with L a linear form (or as a sum of ♯ ( S ) d -powers of vectors). Here we fix an integral variety T ⊊ ℙ m and P ∈ 〈 ν d ( T ) 〉 and study a similar decomposition with S ⊈ T and ♯ ( S ) minimal. For instance, if T is a linear subspace, then we prove that ♯ ( S ) ≥ ♯ ( S ∩ T ) + d + 1 and classify all ( S, P ) such that ♯ ( S ) - ♯ ( S ∩ T ) ≤ 2 d - 1 .