# Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Authors: E. Ballico
Journal: Geometry
 (2013)

---

## Abstract

LetX⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg.


---

## Body

## 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 ℂ (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 𝕂 such that  char (𝕂)=0 (e.g., ℂ), but for homogeneous polynomials we also work over ℝ (see Corollary 3). Let X⊂ℙr be an integral and nondegenerate variety. Fix P∈ℙr. A tangent vector ofX or a tangent vector ofXreg or a smooth tangent vector ofX is a zero-dimensional connected subscheme of X whose support is a smooth point of X, that is, a point of Xreg, and with degree 2. Fix O∈Xreg and let m be the dimension of X at O. The set of all smooth tangent vectors of X with O as its support is parametrized by a projective space of dimension m-1. If 𝕂=ℂ, X is defined over ℝ and O∈Xreg(ℝ), a smooth tangent vector Z⊂X with Zred={O} is said to be real if it is defined over ℝ. A zero-dimensional scheme Z⊂X is said to be curvilinear if for each connected component W of Z either W is a point of X or there is Wred∈Xreg and W is contained in a smooth curve contained in an open neighborhood of Wred in X. A zero-dimensional scheme Z⊂X is said to be smoothable if it is a flat limit of a flat family of finite subsets of X (a curvilinear scheme is smoothable). Fix P∈ℙr. The X-rank rX(P) of P is the minimal cardinality of a finite set S⊂X such that P∈〈S〉, where 〈〉 denote the linear span. The scheme X-rank (or X-cactus rank) zX(P) of P is the minimal degree of a zero-dimensional scheme Z⊂X such that P∈〈Z〉 [16, Definition 5.1, page 135, Definition 5.66, page 198, 31, 12, 10, 11, 17, 18, 8, 9]. If we impose that Z is smoothable (curvilinear, resp.), then we get the smoothable X-rank zX′(P) (curvilinear X-rank zX′′(P), resp.) of [17, 18] for wonderful uses of the scheme X-rank. Let wX(P) be the minimal degree of a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and each connected component of Z is either a point of X or a smooth tangent vector of X (any such Z is curvilinear). We have
(1)zX(P)≤zX′(P)≤zX′′(P)≤wX(P).
Hence to get an upper bound for the integer zX(P), it is sufficient to find an upper bound for the integer wX(P). 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 integersm and d let νd:ℙm→ℙr, r:=(m+dm)-1, denote the order d Veronese embedding of ℙm, that is, the embedding of ℙm given by the 𝕂-vector space of all degree d homogeneous polynomials in m+1 variables.Theorem 1.
Fix integersm≥2 and d≥3. If m≤4, then assume d≥5. Let Xm,d⊂ℙr, r:=(m+dm)-1, be the order d Veronese embedding of ℙm. Set c:=⌈(m+dm)/(m+1)⌉. Let Ω⊂Xm,d be any nonempty open subset of Xm,d. Then there is a disjoint union Z⊂Ω of c tangent vectors such that P∈〈Z〉.Corollary 2.
In the setup of Theorem1, one has wXm,d(P)≤2⌈(m+dm)/(m+1)⌉ for all P∈ℙr.Corollary 3.
LetXm,d(ℝ)⊂ℙr(ℝ), r:=(m+dm)-1, be the order d Veronese embedding of ℙm(ℝ). Fix P∈ℙr(ℝ) and a nonempty open subset U⊆ℙm(ℝ) for the euclidean topology. Then there is S⊂U and for each Q∈S a real tangent vector vQ of ℙm(ℝ) such that #(S)≤⌈(m+dm)/(m+1)⌉ and P∈〈∪Q∈Sνd(vQ)〉.Theorem1 is just a particular case of a general bound on wX(P) (see Theorem 4). We want to point out two features of these results.(i)
The use of an arbitrary nonempty open subsetΩ (U, resp.) of Xm,d (ℙm(ℝ), 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 rX(P) or zX(P) and with supports away from some bad varieties [10, 11, 14, 19]. For instance, in [19] Jelisiejew takes as Ω the image by the Veronese embedding νd 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 integerwX(P).LetX⊂ℙr be an integral and nondegenerate variety. Set m:=dim(X). For each integer b>0 the b-secant variety σb(X)⊆ℙr of X is the closure in ℙr of the union of all linear spaces 〈S〉, where S⊂X is a subset with cardinality b. The set σb(X) is an integral variety of dimension at most min{r,(m+1)b-1}. In many important cases, the integer dim(σb(X)) is known and either dim(σb(X))=min{r,(m+1)b-1} or dim(σb(X))-min{r,(m+1)b-1} is very small [20, 21]. Hence it is usually easy to find an integer c with c-⌈(r+1)/(m+1)⌉ small such that σc(X)=ℙr.Theorem 4.
LetX⊂ℙr be an integral and nondegenerate variety. Let c be the first positive integer such that σc(X)=ℙr. Fix any nonempty open subset U of X and any P∈ℙr. Then one has the following.(a)
There is a disjoint unionZ⊂U of c smooth tangent vectors such that P∈〈Z〉;(b)
wX(P)≤2c.Remark 5.
TakeX and c as in Theorem 4. We have c=rX(O) with O a general element of ℙr. Hence the scheme X-rank of the worst point of ℙr is at most twice the rank of almost all points of ℙr.


## 2. The Proofs

Proof of Theorem4.
Fix a generalS⊂U∩Xreg such that #(S)=c. For each O∈Xreg let TOX⊂ℙr denote the Zariski tangent space of X at O. Since σc(X)=ℙr, Terracini's lemma gives ℙr=〈∪O∈STOX〉 [20, Corollary 1.11]. Hence for each O∈S, there is PO∈TOX such that P∈〈∪O∈SPO〉. Fix O∈S. If PO=O, then let vO be any tangent vector of X at O. Now assume that PO≠O. Since the line L:=〈{PO,O}〉 is contained in TOX, the scheme L∩X contains the tangent vector vO of L at O. Set Z:=∪O∈SvO. Since PO∈〈vO〉 for all O∈S, we have P∈〈Z〉.Proof of Theorem1.
Since eitherm≥5 and d≥3 or d≥5, a theorem of Alexander and Hirschowitz says that c is the first positive integer t such that σt(Xm,d)=ℙr [22–25]. Apply Theorem 4.Proof of Corollary2.
This is a consequence of Theorem1.Proof of Corollary3.
For any subsetA of ℙr(ℝ) we write 〈A〉 for its linear span over ℂ and 〈A〉ℝ for its linear span over ℝ. For each Q∈Xm,d(ℝ), we write TQXm,d(ℂ) for the Zariski tangent space over ℂ and write TQXm,d(ℝ) for the real tangent space (hence both projective spaces have dimension m, the first one over ℂ, the second one over ℝ). We have TQXm,d(ℂ)∩ℙr(ℝ)=TQXm,d(ℝ). The set νd(U) is Zariski dense in Xm,d(ℂ). Hence Terracini's lemma gives the existence of S⊂U such that #(S)=a and 〈∪Q∈νd(S)TQXm,d(ℂ)〉=ℙr(ℂ). Since νd(S)⊂Xm,d(ℝ) and P∈ℙr(ℝ), we get P∈〈∪O∈νd(S)TOXm,d(ℝ)〉ℝ. Hence for each O∈νd(S) there is PO∈TOXm,d(ℝ) such that P∈〈∪O∈νd(S)PO〉ℝ. Continue as in the proof of Theorem 4.



---

*Source: 614195-2013-09-08.xml*