An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables
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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