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

LetX ⊂ Pr be an integral and nondegenerate variety. Let c be the minimal integer such that Pr is the c-secant variety ofX, 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 ofXreg.


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][2][3][4][5][6][7][8][9][10][11][12][13][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  ⊂ P  be an integral and nondegenerate variety.Fix  ∈ P  .A tangent vector of  or a tangent vector of  reg 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  reg , and with degree 2. Fix  ∈  reg 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  − 1.If K = C,  is defined over R and  ∈  reg (R), a smooth tangent vector  ⊂  with  red = {} is said to be real if it is defined over R. A zero-dimensional scheme  ⊂  is said to be curvilinear if for each connected component  of  either  is a point of  or there is  red ∈  reg and  is contained in a smooth curve contained in an open neighborhood of  red 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  ∈ P  .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 zerodimensional 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 ]  : P  → P  ,  := ( +  ) − 1, denote the order  Veronese embedding of P  , that is, the embedding of P  given by the K-vector space of all degree  homogeneous polynomials in  + 1 variables.Corollary 3. Let  , (R) ⊂ P  (R),  := ( +  ) − 1, be the order  Veronese embedding of P  (R).Fix  ∈ P  (R) and a nonempty open subset  ⊆ P  (R) for the euclidean topology.Then there is  ⊂  and for each  ∈  a real tangent vector V  of P  (R) such that #() ≤ ⌈( +  ) /( + 1)⌉ and  ∈ ⟨∪ ∈ ]  (V  )⟩.
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 Ω (, resp.) of  , (P  (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   () or   () 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 ]  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   ().
Remark 5. Take  and  as in Theorem 4. We have  =   () with  a general element of P  .Hence the scheme -rank of the worst point of P  is at most twice the rank of almost all points of P  .If   = , then let V  be any tangent vector of  at .Now assume that   ̸ = .Since the line  := ⟨{  , }⟩ is contained in   , the scheme  ∩  contains the tangent vector V  of  at . Set  := ∪ ∈ V  .Since   ∈ ⟨V  ⟩ for all  ∈ , we have  ∈ ⟨⟩.
Proof of Corollary 2. This is a consequence of Theorem 1.