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 [1–14]. 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
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