Volume 2013 (2013), Article ID 614195, 3 pages
Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties
Department of Mathematics, University of Trento, Povo, 38123 Trento, Italy
Received 3 June 2013; Accepted 9 August 2013
Academic Editor: Anna Fino
Copyright © 2013 E. Ballico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 .
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 positive integers and let , , denote the order Veronese embedding of , that is, the embedding of given by the -vector space of all degree homogeneous polynomials in variables.
Theorem 1. Fix integers and . If , then assume . Let , , be the order Veronese embedding of . Set . Let be any nonempty open subset of . Then there is a disjoint union of tangent vectors such that .
Corollary 2. In the setup of Theorem 1, one has for all .
Corollary 3. Let , , be the order Veronese embedding of . Fix and a nonempty open subset for the euclidean topology. Then there is and for each a real tangent vector of such that and .
Theorem 1 is just a particular case of a general bound on (see Theorem 4). We want to point out two features of these results.(i)The use of an arbitrary nonempty open subset (, resp.) of (, resp.). This is not just to get a formally stronger statement. In many cases, the inductive proofs require the existence of sets (or schemes) bounding or and with supports away from some bad varieties [10, 11, 14, 19]. For instance, in  Jelisiejew takes as the image by the Veronese embedding of the complement of finitely many hyperplanes; he calls it the “open rank.”(ii)We use very particular curvilinear schemes, just disjoint unions of tangent vectors. One should find algorithms to find the support and the direction of tangent vectors needed to compute a good upper bound for the integer .
Let be an integral and nondegenerate variety. Set . For each integer the -secant variety of is the closure in of the union of all linear spaces , where is a subset with cardinality . The set is an integral variety of dimension at most . In many important cases, the integer is known and either or is very small [20, 21]. Hence it is usually easy to find an integer with small such that .
Theorem 4. Let be an integral and nondegenerate variety. Let be the first positive integer such that . Fix any nonempty open subset of and any . Then one has the following.(a)There is a disjoint union of smooth tangent vectors such that ;(b).
Remark 5. Take and as in Theorem 4. We have with a general element of . Hence the scheme -rank of the worst point of is at most twice the rank of almost all points of .
2. The Proofs
Proof of Theorem 4. Fix a general such that . For each let denote the Zariski tangent space of at . Since , Terracini's lemma gives [20, Corollary 1.11]. Hence for each , there is such that . Fix . If , then let be any tangent vector of at . Now assume that . Since the line is contained in , the scheme contains the tangent vector of at . Set . Since for all , we have .
Proof of Corollary 3. For any subset of we write for its linear span over and for its linear span over . For each , we write for the Zariski tangent space over and write for the real tangent space (hence both projective spaces have dimension , the first one over , the second one over . We have . The set is Zariski dense in . Hence Terracini's lemma gives the existence of such that and . Since and , we get . Hence for each there is such that . Continue as in the proof of Theorem 4.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
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