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

Authors: E. Ballico
Journal: Geometry
 (2013)

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## Abstract

Fix integersm≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.)


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## Body

## 1. An Upper Bound for the Symmetric Tensor Rank

Fix positive integersm,d, and an algebraically closed field 𝕂 such that either char(𝕂)=0 or char(𝕂)>d. Let 𝕂[x0,…,xm]d be the 𝕂-vector space of all degree d homogeneous polynomials in m+1 variables. For any f∈𝕂[x0,…,xm]d, the symmetric rank (or the symmetric tensor rank or just the rank) rm,d(f) of f is the minimal integer t such that f=∑i=1t‍ℓid for some ℓi∈𝕂[x0,…,xm]1 [1–5]. Let ρ(m,d) be the maximum of all integers rm,d(f), f∈𝕂[x0,…,xm]d. It is known that ρ(m,f)≤(m+dm)-m [5, Proposition 5.1], but this is a general upper bound for the X-rank with respect to any projective variety X⊂ℙr (although it is sharp in the case m=1, since ρ(1,d)=d for all d by a theorem of Sylvester ([6], [5, Theorem 5.1], and [2, Theorem 23]). In this paper, we prove the following result.Theorem 1.
Fix integersm,d such that m≥5 and d≥3. Then,
(1)ρ(m,d)≤d·⌈(m+dm)(m+1)⌉.Of course, by [5, Proposition 5.1], Theorem 1 is nontrivial only if m≫d. Since either char(𝕂)=0 or char(𝕂)>d, Theorem 1 is equivalent to the following result.Corollary 2.
A symmetricd-tensor in m+1 variables has symmetric tensor rank at most d·⌈(m+dm)/(m+1)⌉.


## 2. The proof

For any subsetS⊂ℙr, let 〈S〉 denote its linear span, that is, the intersection of all hyperplanes containing S, with the convention 〈S〉=ℙr if there is no such a hyperplane. For any integral variety Y⊂ℙr and any integer t>0, the t-secant variety σt(Y) of Y is the closure of the union of all linear spaces 〈S〉, where S⊂Y is the union of any t linearly independent points. Let μd:ℙm→ℙN, N:=(m+dm)-1, denote the Veronese embedding of ℙm, that is, the embedding of ℙm induced by the complete linear system |𝒪ℙm(d)|. Set Xm,d:=μd(ℙm). For any Q∈ℙm, let 2Q denote the first infinitesimal neighborhood of Q in ℙm, that is, the closed subscheme of ℙm with (ℐQ)2 as its ideal sheaf. For any finite set B⊂ℙm set 2B:=∪Q∈B2Q. For any O∈Xm,d, let TOXm,d⊂ℙN denote the Zariski tangent space of Xm,d at O. Notice that Tνd(Q)Xm,d=〈νd(2Q)〉;thatis,Tνd(Q)Xm,d is the minimal linear subspace of ℙN containing the zero-dimensional scheme νd(2Q) of Xm,d. Hence, 〈∪O∈νd(B)TOXm,d〉=〈νd(2B)〉 for any finite B⊂ℙm. Notice that 〈∪O∈νd(B)TOXm,d〉=ℙN if and only if h0(ℙm,ℐ2B(d))=0.Proof of Theorem1.
Setα:=⌈(m+dm)/(m+1)⌉. Since m≥5 and d≥3, a theorem of Alexander and Hirschowitz says that α is the first positive integer t such that σt(Xm,d)=ℙN [7–10]. In characteristic zero, this is equivalent to say that α is the first positive integer t such that 〈∪Q∈BTνd(Q)Xm,d〉=ℙN for a general B⊂ℙm with cardinality t, that is, the first integer t such that h0(ℙm,ℐ2B(d))=0 for the union B⊂ℙm of t general points of ℙm [11, Terracini’s lemma, part 2 of Corollary 1.11]. In a positive characteristic, only one implication holds; that is, if h0(ℙm,ℐ2B(d))=0 for a general B⊂ℙm with ♯(B)=t, then σt(Xm,d)=ℙN [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 A⊂Xm,d such that ♯(A)=α, we have 〈∪O∈ATOXm,d〉=ℙN. Fix a general A⊂Xm,d with ♯(A)=α. Fix any f∈𝕂[x0,…,xm]∖{0}, and let P∈ℙN be the point associated to any λf, λ∈𝕂∖{0}. By the definition of the Veronese embedding, the symmetric tensor rank rm,d(f) is the minimal cardinality of a set S⊂Xm,d such that P∈〈S〉. Since 〈∪O∈ATOXm,d〉=ℙN, for each O∈A, there is PO∈TOXm,d such that P∈〈∪O∈APO〉. If PO∈Xm,d, then it has a symmetric rank 1, while if PO∉Xm,d, then it has symmetric rank d [2, Theorem 32]. Hence, for each O∈A, there is SO⊂Xm,d such that PO∈〈SO〉 and ♯(SO)≤d. Set S:=∪O∈ASO. Since ♯(S)≤dα and P∈〈S〉, we have rm,d(f)≤dα.Remark 3.
Fix any integerb≥⌈(m+dm)/(m+1)⌉ and any A⊂Xm,d such that 〈∪O∈ATOXm,d〉=ℙN and ♯(A)=b. We saw that we may take A as any general subset of Xm,d with cardinality b. Fix any f∈𝕂[x0,…,xm]∖{0}, and let P∈ℙNbe the point associated to f. Our choice of the set A implies P∈〈∪O∈ATOXm,d〉. If (a very big IF) it is possible algorithmically to find PO∈TOXm,d, O∈A, such that P∈〈∪O∈APO〉, then it is possible to find explicitly and algorithmically a set S⊂Xm,d such that ♯(S)≤bd and P∈〈S〉. Indeed, there is an algorithm to find SO⊂Xm,d such that PO∈〈SO〉 and ♯(SO)=d [2, Algorithm 3]. Take S:=∪O∈ASO.



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*Source: 715907-2013-03-14.xml*