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Volume 2013 (2013), Article ID 715907, 2 pages
An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables
Department of Mathematics, University of Trento, 38123 Povo, Italy
Received 22 December 2012; Accepted 15 February 2013
Academic Editor: Michel Planat
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.
Fix integers and . Let be a degree homogeneous polynomial in variables. Here, we prove that is the sum of at most -powers of linear forms (of course, this inequality is nontrivial only if .)
1. An Upper Bound for the Symmetric Tensor Rank
Fix positive integers , and an algebraically closed field such that either or . Let be the -vector space of all degree homogeneous polynomials in variables. For any , the symmetric rank (or the symmetric tensor rank or just the rank) of is the minimal integer such that for some [1–5]. Let be the maximum of all integers , . It is known that [5, Proposition 5.1], but this is a general upper bound for the -rank with respect to any projective variety (although it is sharp in the case , since for all by a theorem of Sylvester (, [5, Theorem 5.1], and [2, Theorem 23]). In this paper, we prove the following result.
Theorem 1. Fix integers such that and . Then,
Corollary 2. A symmetric -tensor in variables has symmetric tensor rank at most .
2. The proof
For any subset , let denote its linear span, that is, the intersection of all hyperplanes containing , with the convention if there is no such a hyperplane. For any integral variety and any integer , the -secant variety of is the closure of the union of all linear spaces , where is the union of any linearly independent points. Let , , denote the Veronese embedding of , that is, the embedding of induced by the complete linear system . Set . For any , let denote the first infinitesimal neighborhood of in , that is, the closed subscheme of with as its ideal sheaf. For any finite set set . For any , let denote the Zariski tangent space of at . Notice that ; is the minimal linear subspace of containing the zero-dimensional scheme of . Hence, for any finite . Notice that if and only if .
Proof of Theorem 1. Set . Since and , a theorem of Alexander and Hirschowitz says that is the first positive integer such that [7–10]. In characteristic zero, this is equivalent to say that is the first positive integer such that for a general with cardinality , that is, the first integer such that for the union of general points of [11, Terracini’s lemma, part 2 of Corollary 1.11]. In a positive characteristic, only one implication holds; that is, if for a general with , then [11, part (1) of Corollary 1.11]. This is exactly the version of Alexander-Hirschowitz theorem proved in arbitrary characteristic by Chandler [9, Theorem 1]. Hence, for a general such that , we have . Fix a general with . Fix any , and let be the point associated to any , . By the definition of the Veronese embedding, the symmetric tensor rank is the minimal cardinality of a set such that . Since , for each , there is such that . If , then it has a symmetric rank , while if , then it has symmetric rank [2, Theorem 32]. Hence, for each , there is such that and . Set . Since and , we have .
Remark 3. Fix any integer and any such that and . We saw that we may take as any general subset of with cardinality . Fix any , and let be the point associated to . Our choice of the set implies . If (a very big IF) it is possible algorithmically to find , , such that , then it is possible to find explicitly and algorithmically a set such that and . Indeed, there is an algorithm to find such that and [2, Algorithm 3]. Take .
The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
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