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