%0 Journal Article
%T Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties
%A E. Ballico
%J Geometry
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/614195
%X Let be an integral and nondegenerate variety. Let be the minimal integer such that is the -secant variety of , that is, the minimal integer such that for a general there is with and , where is the linear span. Here we prove that for every there is a zero-dimensional scheme such that and ; we may take as union of points and tangent vectors of . 1. Introduction There is a huge literature on the rank of tensors, on the symmetric tensor rank of symmetric tensors, and on the Waring decomposition of multivariate polynomials [1¨C14]. Most of the papers are over (or over an algebraically closed field), but real tensors and real polynomials are also quite studied [6, 15]. In this paper we work over an algebraically closed field such that char£¿ (e.g., ), but for homogeneous polynomials we also work over (see Corollary 3). Let be an integral and nondegenerate variety. Fix . A tangent vector of or a tangent vector of or a smooth tangent vector of is a zero-dimensional connected subscheme of whose support is a smooth point of , that is, a point of , and with degree . Fix and let be the dimension of at . The set of all smooth tangent vectors of with as its support is parametrized by a projective space of dimension . If , is defined over and , a smooth tangent vector with is said to be real if it is defined over . A zero-dimensional scheme is said to be curvilinear if for each connected component of either is a point of or there is and is contained in a smooth curve contained in an open neighborhood of in . A zero-dimensional scheme is said to be smoothable if it is a flat limit of a flat family of finite subsets of (a curvilinear scheme is smoothable). Fix . The -rank of is the minimal cardinality of a finite set such that , where denote the linear span. The scheme -rank (or -cactus rank) of is the minimal degree of a zero-dimensional scheme such that [16, Definition 5.1, page 135, Definition 5.66, page 198, 31, 12, 10, 11, 17, 18, 8, 9]. If we impose that is smoothable (curvilinear, resp.), then we get the smoothable -rank (curvilinear -rank , resp.) of [17, 18] for wonderful uses of the scheme -rank. Let be the minimal degree of a zero-dimensional scheme such that and each connected component of is either a point of or a smooth tangent vector of (any such is curvilinear). We have Hence to get an upper bound for the integer , it is sufficient to find an upper bound for the integer . We first state our upper bound in the case of the Veronese varieties (this case corresponds to the decomposition of homogeneous polynomials as a sum of powers of linear forms). For all
%U http://www.hindawi.com/journals/geometry/2013/614195/