Alessandra Bernardi
email: <alessandra.bernardi@unito.it>

J Ér

Ôme Brachat

Bernard Mourrain

A COMPARISON OF DIFFERENT NOTIONS OF RANKS OF SYMMETRIC TENSORS

We introduce various notions of rank for a high order symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.

Introduction

The tensor decomposition problem arises in many applications (see [START_REF] Landsberg | Tensors: Geometry and applications[END_REF] and references therein). Because of many analogies with the matrix Singular Value Decomposition (SVD), this multilinear generalization to high order tensors that we are going to consider, is often called "higher-order singular vale decomposition (HOSVD)" ( [START_REF] Lathauwer | A multilinear singular value decomposition[END_REF]). HOSVD is a linear algebra method often used as a way to recover geometric or intrinsic informations, "hidden" in the tensor data. For a given tensor with a certain structure, this problem consists in finding the minimal decomposition into indecomposable tensors with the same structure. The best known and studied case is the one of completely symmetric tensors (see examples in [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF], [START_REF] Comon | Independent Component Analysis[END_REF], [START_REF] Comon | Symmetric Tensors and Symmetric Tensor Rank[END_REF]), i.e. homogeneous polynomials. The minimum number r of indecomposable symmetric tensors v ⊗d i 's (pure powers of linear forms l i 's) needed to write a given symmetric tensor T of order d (that is a homogeneous polynomial f of degree d) is called the rank r(T ) of T (the rank r(f ) of f ):

T = r i=1 v ⊗d i ; f = r i=1 l d i .
Observe that when d = 2, i.e. when the tensor T is a matrix (i.e. when the homogeneous polynomial is a quadric), this coincides with the standard definition of rank of a matrix. In that case, a tensor decomposition of a symmetric matrix (that can be obtained by SVD computation) of rank r, will allow to write it as a linear combination of r symmetric matrices of rank 1.

From now on, with an abuse of notation, we will denote with "f " both a symmetric tensor and its associate homogeneous polynomial.

From a geometric point of view, saying that a symmetric tensor f has rank r, means that it is in the r-th secant of the Veronese variety in the projective space of polynomials of degree d. The order r σ (f ) of the smallest secant variety to the Veronese variety containing a given f is called the border rank of T and may differs from the rank of f (see Example 2.2).

A first method to decompose a high order symmetric tensor is classically attributed to Sylvester and it works for tensors f ∈ V ⊗d with dim V = 2 (i.e. for binary forms). Such a method (see for a modern reference [START_REF] Comas | On the rank of a binary form[END_REF]) is based on the analysis of the kernel of so-called catalecticant matrices associated to the tensor. This leads to the notion of catalecticant rank r H (f ) of a tensor f , which is also called "differential length" in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][ Definition 5.66,p.198].

Extending the apolarity approach of Sylvester, an algorithm to compute the decomposition and the rank of a symmetric tensor f in any dimension was described in [START_REF] Brachat | Symmetric Tensor Decomposition[END_REF]. The main ingredient of this work is an algebraic characterization of the property of flat extension of a catalecticant matrix. This extension property is not enough to characterize tensors with a given rank, since the underlying scheme associated to the catalecticant matrix extension should also be reduced.

To get a better insight on this difference, we introduce hereafter the notions of extension rank r E 0 (f ) and border extension rank r E (f ) of f , and analyze the main properties.

Another approach leading to a different kind of algorithm is proposed in [START_REF] Bernardi | Computing symmetric rank for symmetric tensors[END_REF] and it is developed for some cases. The idea there, is to classify all the possible ranks of the polynomials belonging to certain secant varieties of Veronese varieties in relation with the structure of the embedded non reduced zero-dimensional schemes whose projective span is contained in that secant variety. In [START_REF] Buczyńska | Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes[END_REF], the authors clarify the structure of the embedded schemes whose span is contained in the secant varieties of the Veronese varieties. Moreover they introduce an algebraic variety, namely the r-th cactus variety K d r . This lead us to the notion of what we will call the border scheme length r sch (f ) of a polynomial f . We will show that this notion is related to the scheme length associated to f defined in ( [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][Definition 5.1, p. 135, Definition 5.66, p. 198]), we will call it the scheme length which is sometimes called the cactus rank of a homogeneous polynomial f (see [START_REF] Ranestad | On the rank of a symmetric form[END_REF] for a first definition of it).

Another notion related to the scheme length and called the smoothable rank r smooth 0 (f ) of a homogeneous polynomial f is also used in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][Definition 5.66, p. 198] or [START_REF] Ranestad | On the rank of a symmetric form[END_REF]. Instead of considering all the schemes of length r apolar to f , one considers only the smoothable schemes, that are the schemes which are the limits of smooth schemes of r simple points. Analogously we can define the border smoothable rank r smooth (f ) of a homogeneous polynomial f , as the smallest r such that f belongs to the closure of the set of tensors of smoothable rank r.

In relation with the "generalized additive decomposition" of a homogeneous polynomial f , there is the so called "length of f ": it was introduced for binary forms in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][Definition 1.30, p. 22], and extended to any form in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][ Definition 5.66,p. 198]. In this paper we will describe a new generalization of the notion of generalized affine decomposition of a homogeneous polynomial f (see Definition 2.16) and study the corresponding generalized rank r G 0 (f ). Again there is a notion of border generalized rank r G (f ).

As in the classical tensor decomposition problem, the decompositions associated to these different notions of rank can be useful to analyze geometric information "hidden" in a high order tensor. The purpose of this paper is to relate all these notions of rank. This will give an algebraic geometric insight to a multilinear algebra concept as HOSVD.

In Corollary 3.9 we will show that the generalized rank, the scheme length and the flat extension rank coincide:

r G 0 (f ) = r sch 0 (f ) = r E 0 (f ).
and hence their respective "border versions":

r G (f ) = r sch (f ) = r E (f ).
We can summarize the relations among the ranks in the following table :   (1)

r H (f ) ≤                          r G (f ) ≤ r G 0 (f ) = = r sch (f ) ≤ r sch 0 (f ) = = r E (f ) ≤ r E 0 (f ) ≥ ≥ r smooth (f ) ≤ r smooth 0 (f ) = r σ (f )                          ≤ r(f ). Let G d,0 r , K d,0
r and E d,0 r be the sets of homogeneous polynomial of degree d in a given number of variables of generalized rank, scheme length and extension rank respectively less than or equal to r and let G d r , K d r and E d r their Zariski closures. The main results of this paper is Theorem 3.7 where we show that

G d,0 r = K d,0 r = E d,0
r , and hence (Corollary 3.8) that

G d r = K d r = E d r .
The paper is organized as follows. After the preliminary Section 1 where we introduce some preliminary material on multilinear algebra and algebraic geometry needed for further developments, we will define, in Section 2, all the definitions of rank that we want to study and for each one of them we will give detailed examples. In Sections 3 we will prove our main results.

1. Preliminaries 1.1. Notations. Let S = K[x] be the graded polynomial ring in the variables x = (x 0 , . . . , x n ) over an algebraically closed field K of characteristic 0. For d ∈ N, let S d be the the vector space spanned by the homogeneous polynomials of degree d in S. We denote by R = K[x] the ring of polynomials in the variables x = (x 1 , . . . , x n ) and by R ≤d the vector space of polynomials in R of degree ≤ d. An ideal I ⊂ S is homogeneous if it can be generated by homogeneous elements.

For f ∈ S d , we denote by f = f (1, x 1 , . . . , x n ) ∈ R ≤d the polynomial obtained by substituting x 0 by 1. This defines a bijection between S d and R ≤d , which depends on the system of coordinates chosen to represent the polynomials. For

f ∈ R, we define f h (x 0 , . . . , x n ) = x deg(f ) 0 f ( x 1
x 0 , . . . , xn x 0 ) and we call it the homogenization of f . A set B of monomials of R is connected to 1 if it contains 1 and if m = 1 ∈ B then there exists 1

≤ i ≤ n and m ′ ∈ B such that m = x i m ′ . For a set B of monomials in R, B + = B ∪ x 1 B ∪ • • • ∪ x n , B.
We denote by P n := P(K n+1 ) the projective space of dimension n. A point in P n which is the class of the non-zero element k = (k 0 , . . . , k n ) ∈ K n+1 modulo the collinearity relation is denoted by [k] = (k 0 : • • • : k n ). An ideal I ⊂ S is homogeneous if it is generated by homogeneous polynomials. For a homogeneous ideal I ⊂ S, the set of points [k] ∈ P n such that ∀f ∈ I, f (k) = 0 is denoted V P n (I). We say that an ideal I ⊂ S is zero-dimensional if V P n (I) is finite and not empty. We say that ζ ∈ V P n (I) is simple if the localization (S/I) m ζ of S/I at the maximal ideal m ζ associated to ζ is of dimension 1 (cf. [START_REF] Atiyah | Introduction to commutative algebra[END_REF]). An ideal I of S is saturated if (I : S 1 ) = I.

We will denote with I d the dedree d part of an ideal I. The Hilbert function associated to I evaluated at d ∈ N is H S/I (d) = dim(S d /I d ). When I is zerodimensional, the Hilbert function becomes equal to a constant r ∈ N for d ≫ 0. When moreover I is saturated, this happens when d ≥ r (see e.g. [START_REF] Gotzmann | Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes[END_REF] for more details).

For a homogeneous ideal I ⊂ S, let I be the ideal of R, generated by the elements f for f ∈ I. We recall that if H S/I (d) = r for d ≫ 0 and if x 0 is a non-zero divisor in S/I, then R/I is a K-vector space of dimension r. Conversely, if Ĩ is an ideal of R such that dim K (R/ Ĩ) = r then the homogeneous ideal

I = {f h | f ∈ Ĩ} is saturated, x 0 is a non-zero
S d /I d ≃ R ≤d /I ≤d . For a point k = (k 0 , . . . , k n ) ∈ K n+1 , we define a corresponding element k(x) ∈ S 1 as k(x) = k 0 x 0 + • • • + k n x n .
The element k(x) is unique, up to a non-zero multiple: it corresponds to a unique element [k(x)] in P(S 1 ). In the following, we will use the same notation k = k(x) to denote either an element of K n+1 or of S 1 . The following product is sometimes called "Bombieri product" or "Sylvester product". Definition 1.2. For all f, g ∈ S d , we define the apolar product on S d as follows:

f, g = |α|=d f α g α d α .
where

f = |α|=d f α d α x α , g = |α|=d g α d α x α , d α = d! α 0 !•••αn! for |α| = α 0 + • • • + α n = d.
It can also be defined on R ≤d in such a way that for all f, g ∈ S d , f , g = f, g (just by replacing x 0 by 1 in the previous formula).

For any vector space E, we denote by E * = Hom K (E, K) its dual space. Notice that the dual S * is an S-module: ∀Λ ∈ S * , ∀p ∈ S, p • Λ : q → Λ(p q).

For any homogeneous polynomial f ∈ S d , we define the element f * ∈ (S d ) * as follows:

∀g ∈ S d , f * (g) = f, g . Similarly, f * ∈ (R ≤d ) * is defined so that ∀g ∈ S d , f * (g) = f , g = f, g .
Let I be an ideal of S. The inverse system I ⊥ of I is the S-submodule of elements of S * that vanish on I, i.e.

I ⊥ = {Λ ∈ S * | ∀f ∈ I, Λ(f ) = 0}. For D ⊂ R * , we define D ⊥ ⊂ R as D ⊥ := {p ∈ R | ∀Λ ∈ D, Λ(p) = 0}.
We check that if D is a R-module, then D ⊥ is an ideal.

When I is a homogeneous ideal, an element in I ⊥ is a sum (not necessarily finite) of elements in (I d ) ⊥ .

Remark 1.3. The dimension of the degree d part of the inverse system of an ideal I ⊂ S is the Hilbert function of S/I in degree d:

H S/I (d) = dim K (I d ) ⊥ = codim(I d ).
We denote by (d α ) |α|=d the basis of (S d ) * that is dual to the standard monomial basis (x β ) |β|=d of S d , more precisely

d α = d α 0 0 • • • d αn n and d α (x β ) = 1 if α = β
and 0 otherwise. An element in (S d ) * is represented by a homogeneous polynomial of degree d in the dual variables d 0 , . . . , d n . It will also be called a dual polynomial.

We remark that and by keeping the terms with positive exponents. This property explains the name of inverse system introduced by F.S. Macaulay [START_REF] Macaulay | The Algebraic Theory of Modular Systems[END_REF]. The dual monomials are also called divided powers in some works, when a structure of ring is given to S * (see e.g. [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][Appendix A]), but this structure is not really needed in the following. It comes from the description of d α in terms of differentials: ∀p ∈ S,

x i • d α = d α 0 • • • d α i-1 i-1 d α i -1 i d α i+1 • • • d αn n if α i > 0
d α (p) = 1 α! ∂ α 0 0 • • • ∂ αn n (p)(0, . . . , 0),
where α! = n i=0 α i !. For D ⊂ S * , we define the inverse system generated by D as the S-module of S * generated by D, that is the vector space spanned by the elements of the form

x α • Λ for α ∈ N n+1 and Λ ∈ D. Example 1.4. The inverse system generated by d 0 d 1 is d 0 d 1 , d 0 , d 1 , 1 . It is a vector space of dimension 4 in K[d 0 , d 1 ].
By extension, the elements of S * can be represented by a formal power series in the variables d 0 , . . . , d n .

By restriction, the elements R * are represented by formal power series in the dual variables d 1 , . . . , d n . The elements of (R ≤t ) * are represented by polynomials of degree ≤ t in the variables d 1 , . . . , d n . The structure of R-module of R * shares the same properties as S * : x i acts as the "inverse" of d i . We define the inverse system spanned by D ⊂ R * as the R-module of R * generated by D.

For a non-zero point k ∈ K n+1 , we define the evaluation

1 d k ∈ (S d ) * at k as 1 d k : S d → K p → p(k)
In the following, we may drop the exponent d to simplify notations when it is implicitly defined.

To describe the dual of zero-dimensional ideals defining points with multiplicities, we need to consider differentials. For k ∈ K n+1 and α = (α 0 , . . . , α n ) ∈ N n+1 , we defined

1 k • ∂ α : S → K p → ∂ α 0 0 • • • ∂ αn n (p)(k).
We extend this definition by linearity, in order to define 1 k • φ(∂) ∈ S * for any polynomial φ(∂) in the differential variables ∂ 0 , . . . , ∂ n . We check that the inverse system generated by 1 k • φ(∂) is the vector space spanned by the elements of the form 1 k • φ ′ (∂) where φ ′ is obtained from φ by possibly several derivations with respect to the differential variables ∂ 0 , . . . , ∂ n . It is a finite dimensional vector space.

This leads to the following result, which characterizes the dual of a zerodimensional (affine) ideal (see e.g. [START_REF] Emsalem | Géométrie des points épais[END_REF] or [START_REF] Elkadi | Introduction à la résolution des systèmes polynomiaux[END_REF][Theorem 7.34, p. 185]).

Theorem 1.5. Suppose that I ⊂ R is such that dim K (R/I) = r < ∞. Then ∀Λ ∈ I ⊥ , there exist distinct points ζ 1 , . . . , ζ s ∈ V K n (I) and differential polynomials φ 1 , . . . , φ s in the variables ∂ 1 , . . . , ∂ n such that Λ = s i=1 1 ζ i • φ i (∂).
As a consequence, we check that the inverse system generated by Λ is the direct sum of the inverse systems D i generated by

1 ζ i • φ i (∂) for i = 1, . . . , s. The sum of the dimensions of these inverse systems is thus ≤ dim K (I ⊥ ) = dim K (R/I) = r. Proposition 1.6. Let Λ = s i=1 1 ζ i • φ i (∂)
, D be the inverse system (or Rmodule) generated by Λ and D i be the inverse system generated by

1 ζ i • φ i (∂) for i = 1, . . . , s. Then D ⊥ = Q i ∩ • • • ∩ Q s where • Q i = D ⊥ i is a primary ideal for the maximal ideal m ζ i defining ζ i , • µ i = dim K (D i ) = dim K (R/Q i ) is the multiplicity of ζ i , • dim R/D ⊥ = s i=1 µ i . Example 1.7. Let us consider the ideal I = (x 2 1 + x 2 -1, x 2 2 -1) of R = K[x 1 , x 2 ]. It defines the points (0, 1), ( √ 2, -1), (- √ 2, -1) ∈ K 2 . An element Λ ∈ I ⊥ can be decomposed as Λ = 1 (0,1) • (a 1 ∂ 1 + b 1 ) + λ 2 1 ( √ 2,1) + λ 3 1 (- √ 2,-1)
where

a 1 , b 1 , λ 2 , λ 3 ∈ K. If a 1 = 0, λ 2 = 0, λ 3 = 0, then the inverse system spanned by Λ is 1 (0,1) • ∂ 1 , 1 (0,1) , 1 ( √ 2,1) , 1 (- √ 2
f = k d 1 + • • • + k d r .
This problem is also called the Generalized Waring problem as it generalizes the problem of Waring in arithmetic [START_REF] Waring | Meditationes algebraicae[END_REF].

In order to find a decomposition of f ∈ S d as a sum of d-th powers of linear forms, we will consider the polynomials which are apolar to f and use the following result.

Lemma 1.10. For all g ∈ S d , k ∈ S 1 with k = k 0 x 0 + • • • + k n x n , k j ∈ K, for j = 0, . . . , n, it turns out that g, k d = g(k), where g(k) = g(k 0 , . . . , k n ).
Proof. By an explicit computation, we have

k d = |α|=d d α n j=0 k α j j n j=0 x α j j . Thus g, k d = |α|=d d α g α n j=0 k α j j = g(k). Thus if f = k d 1 + • • • + k d r with k i ∈ S 1 and if g ∈ S k is such that g(k i ) = 0 for i = 1, . . . , r, then for all h ∈ S d-k we have g h, f = 0.
This shows that the ideal of polynomials vanishing at the points k 1 , . . . , k r ∈ P n is in the set of polynomials apolar to f . It leads us to the following definition (see also [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF] where the same definition is given via an apolar product that differs from our Definition 1.2 only because it is not defined as an inner product but as a product between S d and S d * ). Definition 1.11 (Apolar ideal). Let f ∈ S d . We define the apolar ideal of f as the homogeneous ideal of S generated by S d+1 and by the polynomials g ∈ S i (0

≤ i ≤ d) such that gh, f = 0 for all h ∈ S d-i . It is denoted (f ⊥ ). Example 1.12. For f := x α 0 0 • • • x αn n with α 0 + • • • + α n = d, we have f * = d α -1 d α 0 0 • • • d αn n and (f ⊥ ) = (x α 0 +1 0 , . . . , x αn+1 n ).
Hereafter, we will need the following standard lemma.

Lemma 1.13. For any ideal

I ⊂ S, I d , f = 0 if and only if I ⊂ (f ⊥ ). Proof. Clearly, if I ⊂ (f ⊥ ) then I d ⊂ (f ⊥ ) d so that I d , f = 0.
Let us prove the reverse inclusion. By definition of the apolar ideal J := (f ⊥ ), we have

J i : S k = J i-k , ∀ 0 ≤ i ≤ d, 0 ≤ k ≤ i. We also have I d : S k ⊃ I d-k , ∀ 0 ≤ k ≤ d. The hypothesis I d , f = 0 implies that I d ⊂ J d . We deduce that I i ⊂ J i , ∀ 0 ≤ i ≤ d. Since J d+1 = S d+1 , we have the inclusion I ⊂ J = (f ⊥ ).
The tensor decomposition problem can be reformulated in terms of apolarity as follows via the well known Apolarity Lemma (cf. [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF]Lemma 1.15]). As I ⊂ (f ⊥ ), we have f * ∈ (I d ) ⊥ so that there exists λ 1 , . . . , λ s ∈ K such that

f * = s i=1 λ i 1 w i . This implies that f = s i=1 λ i w d i = s i=1
(λ

1 d i w i ) d
and f has a decomposition of size ≤ s ≤ r.

Ranks of symmetric tensors

In this section we introduce all the different notions of rank of a homogeneous polynomial f ∈ S d , that we will use all along the paper.

2.1.

Rank and border rank. The following definition is nowadays a classical one, see e.g. [START_REF] Landsberg | Tensors: Geometry and applications[END_REF] and references therein.

σ 0,d r := {[f ] ∈ P(S d )| ∃ k 1 , . . . , k s ∈ S 1 with s ≤ r s.t. f = k d 1 + • • • + k d s }. For any f ∈ S d , the minimal r such that [f ] ∈ σ 0,d r is called the rank of f and denoted r(f ). Example 2.2. Let us describe a decomposition of the monomial f := x α 0 0 • • • x αn n with α 0 + • • • + α n = d of
minimal size, which yields to its rank. We consider the ideal

I ǫ := (x α 1 +1 1 -ǫ α 1 +1 x α 1 +1 0 , x α 2 +1 2 -ǫ α 2 +1 x α 2 +1 0 , . . . , x αn+1 n -ǫ αn+1 x αn+1 0 ) for some ǫ ∈ K \ {0}. It is defining (α 1 + 1) • • • (α n + 1)
simple points (which i th coordinates are ǫ times the (α i + 1)-roots of unity). Let us consider the element Λ of S * defined as follows:

Λ := 1 (α 1 + 1) • • • (α n + 1) d α α 1 k 1 =0 • • • αn kn=0 ǫ α 0 -d ζ k 1 1 • • • ζ kn n 1 (1,ǫζ k 1
where ζ i is a primitive (α i + 1)-th root of unity for i = 1, . . . , n. Then for any monomial

x β = x β 0 0 • • • x βn n , we have Λ(x β ) = 1 (α 1 + 1) • • • (α n + 1) d α α 1 k 1 =0 • • • αn kn=0 ǫ α 0 -d+β 1 +•••+βn ζ (β 1 +1) k 0 1 • • • ζ (βn+1) kn n = 1 ( d α ) ǫ ρ(l 1 ,...,ln)+α 0 -d if ∀i = 1, . . . , n, ∃l i ∈ N + , β i + 1 = l i (α i + 1) 0 otherwise.
where ρ(l 1 , . . . ,

l n ) = l 1 (α 1 + 1) + • • • + l n (α n + 1) -n. Its minimal value on N n + is ρ(1, . . . , 1) = α 1 + • • • + α n = d -α 0 .
The previous computation shows that

(3) Λ |S d = 1 d α l∈N n + |d-α 0 ≤ρ(l)≤d ǫ α 0 +ρ(l)-d d d-ρ(l) 0 d l 1 (α 1 +1)-1 1 • • • d ln(αn+1)-1 n .
Suppose that α 0 = min i=0,...,d α i . Then the ideal I ǫ is included in (f ⊥ ) = (x α 0 +1 0 , . . . , x αn+1 n ). By Proposition 1.14, we deduce that f has a decomposition of size n 0 (α i +1) min i (α i +1) . As ρ(l) ≤ d implies l = (1, . . . , 1), we have

Λ |S d = 1 d α d α 0 0 d α 1 1 • • • d αn n = f *
which gives the decomposition of f * . The corresponding decomposition of f in terms of d th -powers of linear forms is

x α 0 0 • • • x αn n = 1 (α 1 + 1) • • • (α n + 1) d α × α 1 k 1 =0 • • • αn kn=0 ǫ d-α 0 ζ k 1 1 • • • ζ kn n x 0 + ǫζ k 1 1 x 1 + • • • + ǫζ kn n x n d .
It can be proved that this decomposition has a minimal size (see [START_REF] Carlini | The Solution to Waring's Problem for Monomials[END_REF], [START_REF] Carlini | The solution to the Waring problem for monomials and the sum of coprime monomials[END_REF]), so that we have

r(x α 0 0 • • • x αn n ) = n 0 (α i + 1) min i (α i + 1)
.

This example also shows that the decomposition is not unique, since ǫ is any non-zero constant.

For more details on rank of monomials see also [START_REF] Catalisano | On the ideals of secant varieties to certain rational varieties[END_REF], [START_REF] Ranestad | On the rank of a symmetric form[END_REF] and [START_REF] Buczyńska | Waring decompositions of monomials[END_REF]; the example above was also shown with different approach in [10, §2] and in [START_REF] Carlini | The Solution to Waring's Problem for Monomials[END_REF]Corollary 3.8].

Definition 2.3 (Border rank). The Zariski closure of σ 0,d r ⊂ P(S d ), also known as the r th secant variety of the Veronese variety of S d , is denoted σ d r . The minimal r such that [f ] ∈ σ d r is called the border rank of f and we denote it r σ (f ) (cf. [START_REF] Bürgisser | Algebraic complexity theory[END_REF][START_REF] Comon | Symmetric Tensors and Symmetric Tensor Rank[END_REF][START_REF] Berge | Typical Rank and INDSCAL dimensionality for Symmetric Three-Way Arrays of order Ix2x2 or Ix3x3[END_REF]).

Example 2.4. Consider again

f := x α 0 0 • • • x αn n with α 0 + • • • + α n = d. Suppose now that α 0 = max i α i . Then the decomposition (3) is of the form f * ǫ = f * + 1 d α l∈N n + |d-α 0 <ρ(l)≤d ǫ α 0 +ρ(l)-d d d-ρ(l) 0 d l 1 (α 1 +1)-1 1 • • • d ln(αn+1)-1 n ,
with possibly some terms in the sum which involves positive powers of ǫ. This shows that lim ǫ→0 f * ǫ = f * . As f * ǫ ∈ I ⊥ ǫ and I ǫ is defining simple points, the rank of f ǫ is ≤

(α i +1) max i (α i +1) .
We deduce that the border rank of f is less than n i=0 (α i +1) max i (α i +1) . In [START_REF] Landsberg | On the ranks and border ranks of symmetric tensors[END_REF] it is shown that if max α i is equal to the sum of all the others α i 's then such a bound is actually sharp. Consider e.g. f = x 0 x d-1 1 (for d > 2). This is the first well known case where the rank and border rank are different: from Example 2.2 we get that r(f ) = d, while here we have just seen that r σ (f ) = 2 (see also [START_REF] Comas | On the rank of a binary form[END_REF], [START_REF] Bernardi | Computing symmetric rank for symmetric tensors[END_REF], [START_REF] Landsberg | On the ranks and border ranks of symmetric tensors[END_REF]).

Smoothable rank. Let Hilb red

r (P n ) be the set of schemes of length r which are the limit of smooth schemes of r points, and let us consider the two following definitions according e.g. to [START_REF] Ranestad | On the rank of a symmetric form[END_REF] and [START_REF] Bernardi | On the cactus rank of cubic forms[END_REF]. Remark 2.7. In [26, Lemma 5.17] it is shown that S d,0 r ⊂ σ d r . This proves that r σ (f ) ≤ r smooth 0 (f ).

The following example is a personal communication from W. Buczyńska and J. Buczyński ( [8]). It shows that strictly inequalities can occur.

Example 2.8 ([8]

). The following polynomial has border rank ≤ 5 but smoothable rank ≥ 6: f = x 2 0 x 2 + 6x 2 1 x 3 -3 (x 0 + x 1 ) 2 x 4 . One can easily check that the following polynomial

f ǫ = (x 0 + ǫx 2 ) 3 + 6(x 1 + ǫx 3 ) 3 -3(x 0 + x 1 + ǫx 4 ) 3 + 3(x 0 + 2 x 1 ) 3 -(x 0 + 3x 1 ) 3
has rank 5 for ǫ > 0, and that lim ǫ→0

1 3ǫ f ǫ = f . Therefore r σ (f ) ≤ 5.
An explicit computation of (f ⊥ ) yields to the following Hilbert function for H R/(f ⊥ ) = [1, 5, 5, 1, 0, . . .]. Let us prove, by contradiction, that there is no saturated ideal I ⊂ (f ⊥ ) of degree ≤ 5. Suppose on the contrary that I is such an ideal. Then 5, 5, 5, . . .]. This shows that I 1 = {0} and that I 2 = (f ⊥ ) 2 . As I is saturated, I 2 : (x 0 , . . . , x 4 ) = I 1 = {0} since H R/(f ⊥ ) (1) = 5. But an explicit computation of ((f ⊥ ) 2 : (x 0 , . . . , x 4 )) gives x 2 , x 3 , x 4 . We obtain a contradiction, so that there is no saturated ideal of degree ≤ 5 such that I ⊂ (f ⊥ ). Consequently, r smooth 0 (f ) ≥ 6 so that r σ (f ) < r smooth 0 (f ). In the introduction we called r smooth (f ) the border smoothable rank of f . 2.3. Catalecticant rank. The apolar ideal (f ⊥ ) can also be defined via the kernel of the following operators. Let us recall the following standard definition. Definition 2.10 (Catalecticant). Given a homogeneous polynomial f ∈ S d and a positive integer k such that k ≤ d, the Catalecticant of order k of f , denoted by

H R/I (n) ≥ H R/(f ⊥ ) (n) for all n ∈ N. As H R/I (n) is an in- creasing function of n ∈ N with H R/(f ⊥ ) (n) ≤ H R/I (n) ≤ 5, we deduce that H R/I = [1,
H k,d-k f *
, is the application:

H k,d-k f * : S k → (S d-k ) * p → p • f * .
Its matrix in the monomial basis {x α } |α|=k of S k and in the basis

{ d-k β -1 d β } |β|=d-k of (S d-k ) * is denoted H k,d-k f * . By construction, ker H k,d-k f * is the component (f ⊥ ) k of degree k of the apolar ideal (f ⊥ ) of f .
Given two families of monomials B ⊂ S k and B ′ ⊂ S d-k , we denote by

H B ′ ,B f * the "restriction" of H k,d-k f *
from the vector space spanned by B to the dual of the vector space spanned by B ′ . Remark 2.11. By symmetry of the apolar product, we have

H k,d-k f * = t H d-k,k f * via the identification S d-i ≃ (S d-i ) * . In terms of matrices, we have H k,d-k f * = t H d-k,k f * . orsalutalo da parte mia H B ′ ,B f * = t H B,B ′ f * for all families of monomials B ⊂ S k and B ′ ⊂ S d-k . Definition 2.12 (Catalecticant rank). Let f ∈ S d . The maximal rank of the operators H k,d-k f * , for 0 ≤ k ≤ d, is called the catalecticant rank of f and it is denoted r H (f ).
This rank was already introduced in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][ Definition 5.66,p.198] where it was called "the differential length of f " and denoted by ldiff(f ). Definition 2.13. Given an integer i ≤ d ∈ N t and r ∈ N, we define the variety Γ i,d-i r ⊂ P(S d ) as:

Γ i,d-i r := {[f ] ∈ P(S d )| rank (H i,d-i f * ) = rank (H d-i,i f * ) ≤ r}. Remark 2.14. The set Γ i,d-i r ⊂ P(S d
) is the algebraic variety defined by the minors (r + 1) × (r + 1) of the catalecticant matrices H i,d-i f * (or H d-i,i f * ). These minors give not necessary reduced equations but they represents in P(S d ) the variety that is the union of linear spaces spanned by the images of the divisors (hypersurfaces in P(S 1 )) of degree r on the Veronese ν d (P(S 1 )) (see e.g. [START_REF] Bernardi | Computing symmetric rank for symmetric tensors[END_REF] and [START_REF] Gherardelli | Observations about a class of determinantal varieties[END_REF]). If i = 1, such a variety is known as the "subspace variety in

P(S d )" Sub r (S d (V )) := P{f ∈ S d (V )|∃W ⊂ V, dim(W ) = r, f ∈ S d (W )}.
For a generic i, it can be geometrically obtained by intersecting P(S d ) with the r-th secant variety of the Segre variety of P(S a ) × P(S d-a ) (for a better description of subspace varieties see [27, §17.4]).

Example 2.15.

For a monomial f = x α 0 0 • • • x αn n with α 0 + • • • + α n = d, the apolar ideal of f is J := (f ⊥ ) = (x α 0 +1 0 , . . . , x α n+1 n ). By Remark 1.3, the rank of H i,d-i f * is the dimension of S i /J i that is the coefficient of t i in h(t) = n i=0 (1 + t + • • • + t α i ).
The maximum value of these coefficients which is the catalecticant rank is reached for the coefficients of the closest degree to 1 2

(α 1 +• • •+α n ) (it is proved in [33]
that the polynomial h(t) is symmetric unimodal, which means that its coefficients are increasing up to the median degree(s) and then decreasing symmetrically). The exact value of the maximum is not known but asymptotic equivalents are known in some cases, see e.g. [19, p. 234-240].

Consider for instance the monomial f = x 0 x 2 1 x 2 2 . The previous computation yields to the following Hilbert series for the apolar ideal:

H S/(f ⊥ ) (t) = (1 + t)(1 + t + t 2 ) 2 = 1 + 3 t + 5 t 2 + 5 t 3 + 3 t 4 + t 5 .
This shows that rank H 1,4 f * = rank H 4,1 f * = 3, rank H 2,3 f * = rank H 3,2 f * = 5 and thus that r H (f ) = 5.

According to Example 2.4, the border rank of f = x 0 x 2 1 x 2 2 is (1 + 1)(2 + 1) = 6, which shows that the border rank of f is strictly bigger that its Catalecticant rank.

In [29, Theorems 1.2.3 and 4.2.7], it is shown that Γ 2,3 5 (P 2 ) has codimension 5 in P(S 5 ) while the secant variety σ 5 5 (P 2 ) has codimension 6. Therefore a generic element of Γ 2,3 5 (P 2 ) has border rank strictly bigger than 5.

2.4.

Generalized rank and border generalized rank.

Definition 2.16. A generalized affine decomposition of size r of f ∈ S d is a decomposition of the form

f * = m i=1 1 ζ i • φ i (∂) on R ≤d
where ζ i ∈ K n and φ i (∂) are differential polynomials, such that the dimension of the inverse systems spanned by m i=1

1 ζ i • φ i (∂) is r. Notice that the inverse system generated by m i=1 1 ζ i • φ i (∂)
is the direct sum of the inverse systems generated by 1 ζ i • φ i (∂) for i = 1, . . . , m. The inverse system generated by 1 ζ i • φ i (∂) is the vector space spanned by the elements

1 ζ i • ∂ α 1 ∂ 1 • • • ∂ αn ∂n φ i (∂) for all α = (α 1 , . . . , α n ) ∈ N n .
This decomposition generalizes the (Waring) decomposition of Definition 1.9, since when φ i (∂) = λ i ∈ K are constant polynomials, we have the decomposition

f * = m i=1 λ i 1 ζ i iff f = m i=1 λ i (1 + ζ i,1 x 1 + • • • ζ i,n x n ) d .
Definition 2.17 (Generalized rank). Given two integers r and d, we define G d,0 r ⊂ P(S d ) by:

G d,0 r := [g]∈P GL(n+1) {[f ] ∈ P(S d ) | g•f * has a generalized affine decomposition of size ≤ r}.
The smallest r such that

[f ] ∈ G d,0
r is called the generalized rank of f and it is denoted r G 0 (f ).

Example 2.18. The polynomial f = x 3 y + y 3 z defines an inverse system of dimension 4 obtained as

1 (1,0,0) , 1 (1,0,0) ∂ y , 1 (0,1,0) , 1 (0,1,0) ∂ z , therefore r G 0 (f ) = 4.
Moreover, we are in a case of a polynomial of border rank 4 and rank 7 (as described in [START_REF] Bernardi | Computing symmetric rank for symmetric tensors[END_REF]Theorem 44]). In this case

r G 0 (f ) = 4 = r σ (f ) < r(f ) = 7. Example 2.19. For a monomial f = x α 0 0 • • • x αn n with α 0 + • • • + α n = d, we have f * = 1 d! 1 (1,0,...,0) • ∂ α 1 1 • • • ∂ αn n . The inverse system spanned by 1 (1,0,...,0) •∂ α 1 1 • • • ∂ αn n is of dimension (α 1 +1)ו • •× (α n +1).
Assuming that α 0 = max i α i , the previous decomposition is a generalized decomposition of minimal size (according to Corollary 3.9 and Example 2.28). Therefore we have

r G 0 (x α 0 0 • • • x αn n ) = n i=0 (α i + 1) max i (α i + 1)
.

Notice that [f ] ∈ G d,0 r iff there exists a change of coordinates such that in the new set of coordinates f has a generalized affine decomposition of size ≤ r.

This notion of generalized affine decomposition and of generalized rank is related to the generalized additive decomposition introduced in [26, Definition 1.30, p. 22] for binary forms, called "the length of f " and denoted l(f ). However, the extension to forms in more variables proposed in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF]Definition 5.66,p. 198] does not correspond to the one we propose, in fact it corresponds to the border rank. For binary forms, the border rank and the generalized rank coincide as we will see in the sequel.

:= {[f ] ∈ P(S d )| ∃ u ∈ S 1 \ {0}, ∃ [ f ] ∈ Γ m,m ′ r with m = max{r, ⌈ d 2 ⌉}, m ′ = max{r -1, ⌊ d 2 ⌋} s.t. u m+m ′ -d • f * = f * }. By definition, m+ m ′ ≥ ⌈ d 2 ⌉+ ⌊ d 2 ⌋ = d. Moreover, if d ≥ 2r -1 then E d r = Γ m,m ′ r since m = ⌈ d 2 ⌉, m ′ = ⌊ d 2 ⌋ and m + m ′ -d = 0. Definition 2.22 (Flat extension rank). The smallest r such that [f ] ∈ E d,0 r is called the flat extension rank of f and it is denoted r E 0 (f ). A [ f ] ∈ Γ m,m ′ r such that ∃ u ∈ S 1 \{0} with u m+m ′ -d • f * = f * and rank H m,m ′ f * = r is called a flat extension of f ∈ S d of rank r. Example 2.23. For a monomial f = x α 0 0 • • • x αn n with α 0 + • • • + α n = d, the element f * = 1 d! 1 (1,0,...,0) • ∂ α 1 1 • • • ∂ αn n ∈ S * ,
defines, by restriction, a Hankel operator H r,r-1 f * from S r to (S r-1 ) * where r = (α 1 + 1) • • • (α n + 1). We check that the image of H r,r-1 f * is the vector space of (S r-1 ) * spanned by 1 (1,0,...,0)

• ∂ β 1 1 • • • ∂ βn n for 0 ≤ β i ≤ α i . Thus H r,r-1 f * is of rank r.
According to Example 2.19 and Theorem 3.5, f is a flat extension of f of minimal rank when α 0 = max i α i . We deduce that

r E 0 (x α 0 0 • • • x αn n ) = n i=0 (α i + 1) max i (α i + 1)
.

Notice that [f ] ∈ E d,0 r iff there exists a change of coordinates such that after this change of coordinates we have u = x 0 so that f * ∈ (R ≤m+m ′ ) * is such that

• f * |R ≤d = f * and • rank H m,m ′ f * ≤ r.
In other words, [f ] ∈ E d,0 r iff after a change of coordinates, f * ∈ (R ≤d ) * can be extended to a linear form f * ∈ (R ≤m+m ′ ) * with rank H m,m ′ f * ≤ r. We will see hereafter that we can choose a generic change of coordinates.

A simple way to characterize a flat extension of a given rank is given by the following result. Theorem 2.24 ([30], [START_REF] Bernardi | Multihomogeneous polynomial decomposition using moment matrices[END_REF], [START_REF]Moment Matrices and Applications[END_REF]). Let M, M ′ be sets of R, B ⊂ M and B ′ ⊂ M ′ be two monomial sets of size r connected to 2.6. Scheme length and border scheme length. We recall that Hilb s (P n ) is the set of 0-dimensional schemes Z of s points (counted with multiplicity). It can be identified with the set of homogeneous saturated ideals I ⊂ S such that the algebra S/I has a constant Hilbert polynomial equal to s. multiplicity s ≤ r. We denote by I ⊂ S the homogenization of I with respect to x 0 . Then, I ∈ Hilb s (P n ). As f * = Λ on R ≤d , we have: f * ∈ (I d ) ⊥ , which proves the inclusion. Lemma 3.2. Given integers r, d and i such that 0 ≤ i ≤ d, we have K d,0 r ⊂ Γ i,d-i r .

1 such that M • M ′ contains B + • B ′+ . If Λ ∈ M • M ′ * is such that rank H B,B ′ Λ = rank H M,M ′ Λ = r, then there exists an extension Λ ∈ R * of Λ such that rank H Λ = r. Moreover, we have ker H Λ = (H M,M ′ Λ ).
Proof. Let us prove that for all f ∈ S d such that I d , f = 0 with I ∈ Hilb s (P n ) and s ≤ r, we have: rank(H i,d-i f * ) ≤ s ≤ r for all 0 ≤ i ≤ d. By Lemma 1.13, we have I ⊂ J := (f ⊥ ) so that

I i ⊂ J i for 0 ≤ i ≤ d.
As the Hilbert function of a saturated ideal I ∈ Hilb s (P n ) is increasing until degree s and then it is constantly equal to s, we have

dim S i /I i ≤ s, ∀ 0 ≤ i ≤ d.
By the above inclusion, this implies that dim S i /J i ≤ s, ∀ 0 ≤ i ≤ d. Proof. Let ≻ be the lexicographic ordering such that x 0 ≻ • • • ≻ x n . By [START_REF] Eisenbud | Commutative Algebra with a view toward Algebraic Geometry[END_REF][Theorem 15.20, p. 351], after a generic change of coordinates, the initial J of the ideal I = (E) for ≻ is Borel fixed. That is, if x i x α ∈ J then x j x α ∈ J for j > i.

As J i := ker H i,d-i f * , we deduce that rank H i,d-i f * ≤ s, ∀ 0 ≤ i ≤ d. Consequently as s ≤ r, f ∈ Γ i,d-i
To prove that there exists a subset B of monomials of degree d -1 such that x 0 B is a basis of S d /I d , we show that

J d + x 0 S d-1 = S d . Let J ′d = (J d + x 0 S d-1 )/x 0 S d-1 , S ′d = S d /x 0 S d-1 = K[x 1 , . . . ,
x n ] and L = (J : x 0 ). Then we have the exact sequence

0 → S d-1 /L d-1 µx 0 -→ S d /J d → S ′d /J ′d ,
where µ x 0 is the multiplication by x 0 . Let us denote by s k = dim S k and q(k) = s k -r for k ∈ N. Suppose that dim S ′d /J ′d > 0, then dim L d-1 > s d-1 -r = q(d -1). As d ≥ r and r is the Gotzmann regularity of q, by [25, (2.10), p. 66] we have dim S 1 L d-1 > q(d). As J is Borel fix, i.e. x 0 p ∈ J implies x i p ∈ J for i ≥ 0, we have

S 1 L d-1 ⊂ J, so that dim J d ≥ dim S 1 L d-1 > q(d) = s d -r.
This implies that dim S d /J d = dim S d /I d = dim S d /E < r, which contradicts the hypothesis on E. Thus J d + x 0 S d-1 = S d . Let B ′ be the complementary of J d in the set of monomials of degree d. The sum S d = J d + x 0 S d-1 shows that B ′ = x 0 B for some subset B of monomials of degree d -1.

As J d is Borel fix and different from S d , its complementary B ′ contains x d 0 . Similarly we check that if

x α 0 0 • • • x αn n ∈ B ′ with α 1 = • • • = α k-1 = 0 and α k = 0 then x α 0 +1 0 x α k -1 k x α k+1 k+1 • • • x αn n ∈ B ′ . This shows that B ′ = B is connected to 1. The equality K d r = Γ i,d-i
G d,0 r = G d r = K d,0 r = K d r = E d,0 r = E d r = Γ i,d-i r .
Proof. Let d ≥ 2r and r ≤ i ≤ d -r. We first prove the following inclusion:

(4) Γ i,d-i r ⊂ G d,0 r . Let us fix [f ] ∈ Γ i,d-i r for an integer r ≤ i ≤ d -r. Let us denote E := Ker(H i,d-i f * ) and F := Ker(H d-i,i
f * ) and k ≤ r the rank of H i,d-i f * . We recall that:

H i,d-i f * = t H d-i,i f * .
The quotients S i /E and S d-i /F are thus of dimension k. As k ≤ r ≤ i and k ≤ r ≤ d -i, by Lemma 3.4 and a generic change of coordinates we may assume that there exists a family B (resp. B ′ ) of k monomials of S i-1 (resp. S d-i-1 ) such that x 0 B (resp. x 0 B ′ ) is a basis of S i /E (resp. S d-i /F ) and that

B ⊂ R ≤i-1 (resp. B ′ ⊂ R ≤d-i-1 ) are connected to 1. Notice then that H B ′ ,B f * = H x 0 B ′ ,x 0 B f * is an invertible matrix of size k × k. As the monomials of B are in R ≤i-1 (resp. R ≤d-i-1 ), the monomials of B + (resp. B ′+ ) are in the set M (resp. M ′ ) of monomials of degree ≤ i (resp. ≤ d -i) and B + • B ′+ ⊂ M • M ′ . Moreover, we have rank H B ′ ,B f * = rank H x 0 B ′ ,x 0 B f * = rank H M ′ ,M f * = rank H i,d-i f * = k.
By Theorem 2.24, there exists a linear form Λ ∈ R * which extends f * such that dim K (R/I Λ ) = r where I Λ = ker H Λ . By Theorem 1.5, as Λ ∈ I ⊥ Λ , there exists

ζ 1 , . . . , ζ m ∈ K n , differential polynomials φ 1 , . . . , φ m ∈ K[∂ 1 , . . . , ∂ n ] such that Λ = i=1,...,m 1 ζ i • φ i (∂).
As the inverse system spanned by Λ is included in (I Λ ) ⊥ which is of dimension s and as Λ coincides with f * on R ≤d , f has a generalized affine decomposition of size ≤ s ≤ r and [f ] ∈ G d r . This proves that Γ i,d-i r ⊂ G d,0 r . Using Lemmas 3.1 and 3.2, we have

(5) Γ i,d-i r ⊂ G d,0 r ⊂ K d,0 r ⊂ Γ i,d-i r . By definition, E d,0 r = Γ r,d-r r when d ≥ 2 r. As Γ i,d-i r is closed, the previous inclusions show that G d,0 r = G d r = K d,0 r = K d r = E d,0 r = E d r = Γ i,d-i r .
To analyze the relationship between the sets G d r , E d r , K d r for general r, d ∈ N, we need the following lemma. Let u ∈ S 1 be a linear form such that (I : u) = I. By a change of coordinates, we can assume that u = x 0 .

We denote by I ⊂ R the dehomogenization of I ⊂ S (setting x 0 = 1). As I ∈ Hilb r (P n ) and (I : x 0 ) = I, the quotient algebra R/I is a K-vector space of dimension r. By the following natural isomorphism:

1 ζ i • φ ′ i (∂)
where φ ′ i is obtained from φ i by derivation. As φ ′ i is obtained from φ i by derivation, the inverse system spanned by

1 ζ i •φ ′ i (∂)
r H (f ) ≤ r G (f ) = r sch (f ) = r E (f ) ≤ ≤ r G 0 (f ) = r sch 0 (f ) = r E 0 (f ) ≤ r(f ).
Proof. Form Corollary 3.3 we get that r H (f ) ≤ r sch (f ) ≤ r sch 0 (f ). By definitions of G d r , K d r and E d r we obviously have that r G (f ) ≤ r G 0 (f ), r sch (f ) ≤ r sch 0 (f ), r E (f ) ≤ r E 0 (f ). Finally Theorem 3.7 and Corollary 3.8 end the proof. Remark 3.10. Obviously S d r ⊂ K d r and S d,0 r ⊂ K d,0 r . This justify r H (f ) ≤ r G (f ) = r sch (f ) = r E (f ) ≤ r smooth (f ) and r G 0 (f ) = r sch 0 (f ) = r E 0 (f ) ≤ r smooth 0 (f ). Now, in order to complete table [START_REF] Alexander | Generic hypersurface singularities[END_REF], it is sufficient to use Remarks 2.7 and 2.9. Observe that in Example 2.8 the condition d ≥ 2r is not satisfied. Since, by Corollary 3.9, we have that r G 0 (f ) = r sch 0 (f ), then Example 2.8 gives also an example of a polynomial f such that r G (f ) = r sch (f ) = r E (f ) < r G 0 (f ) = r sch 0 (f ) = r E 0 (f ).

  divisor in S/I and H S/I (d) = r for d ≥ r. Remark 1.1. If I is a saturated ideal of S and (I : x 0 ) = I, then we have the natural isomorphism for d ∈ N:

  and 0 otherwise. More generally, for any Λ ∈ (S d ) * represented by a dual polynomial of degree d, we have that x i • Λ is either 0 or a dual polynomial of degree d -1. It is formally obtained by multiplying by d -1 i

Proposition 1 . 1 g

 11 [START_REF] Carlini | The solution to the Waring problem for monomials and the sum of coprime monomials[END_REF]. A symmetric tensor f ∈ S d has a decomposition of size s ≤ r iff there exits an ideal I ⊂ S such that (a) I ⊂ (f ⊥ ), (b) I is saturated, zero dimensional, of degree ≤ r, (c) I is defining simple points.Proof. Suppose that f has a decomposition of size ≤ r: f = s 1 w d i where w i ∈ S 1 -{0} and s ≤ r. Then consider the homogeneous ideal I of polynomials vanishing at the points [w i ] ∈ P n , i = 1, . . . , s. By construction, for all g ∈ I d , (w i ) = 0 so that I is a saturated ideal, defining s (≤ r) simple points and with I ⊂ (f ⊥ ).Conversely, suppose that I is an ideal of S satisfying (a), (b), (c). Let us denote by [w 1 ], . . . , [w s ] the simple points of P n defined by I and by w 1 , . . . , w s corresponding elements in S 1 . Then by Lemma 1.8, (I d ) ⊥ is spanned by 1 w 1 , . . . , 1 ws .

Definition 2 . 1 (

 21 Rank). Let σ 0,d r ⊂ P(S d ) be the set of projective classes of homogeneous polynomials defined by

Definition 2 . 5 .

 25 For any integers r and d, we define S r d ⊂ P(S d,0 ), as the setS d,0 r := {[f ] ∈ P(S d )| ∃ s ≤ r, ∃ I ∈ Hilb red r (P n ), I d , f = 0}. This leads to the following definition.

Definition 2 . 6 (

 26 Smoothable rank). The smallest r such that [f ] ∈ S d,0 r is called the smoothable rank of f and it is denoted r smooth 0 (f ).

Remark 2 . 9 .

 29 If we indicate with r smooth (f ) the smallest r such that [f ] ∈ S d,0 r := S d r , then we can observe that S d r = σ d r . Obviously σ d r ⊂ S d r . The other inclusion follows from Remark 2.7. This shows that r σ (f ) = r smooth (f ).

Definition 2 . 20 ( 5 .

 2205 Border generalized rank). Given two integers r and d, we define G d r ⊂ P(S d ) to be the Zariski closure of G d,0 r defined above. The smallest r such that [f ] ∈ G d r is called the border generalized rank of f and it is denoted r G (f ). 2.Flat extension rank and border flat extension rank. We describe here a new notion of rank based the property of extension of bounded rank of the catalecticant matrices. Definition 2.21. For any integers r and d, we define E d,0 r ⊂ P(S d ), as the set E d,0 r

Definition 2 .

 2 25 (border flat extension rank). For any integers r and d, we define E d r ⊂ P(S d ), as the Zariski closure of set E d,0 r defined above and the smallest r such that [f ] ∈ E d r is called the border flat extension rank of f and it is denoted r E (f ).

r.Corollary 3 . 3 .

 33 and K d,0 r ⊂ Γ i,d-i r For any homogeneous polynomial f we have that r H (f ) ≤ r sch (f ) ≤ r sch 0 (f ).Proof. By Lemma 3.2 we have thatK d,0 r ⊂ Γ i,d-i r . Now since Γ i,d-i r is closed by definition, we get that K d,0 r ⊂ K d,0 r = K d r ⊂ Γ i,d-i r that implies that r H (f ) ≤ r sch (f ) ≤ r sch 0 (f ).Lemma 3.4. Let d ≥ r and E ⊂ S d such that S d /E is of dimension r. Then for a generic change of coordinates g ∈ P GL(n + 1), S d /g • E has a monomial basis of the form x 0 B with B ⊂ S d-1 . Moreover, B is connected to 1.

r

  for d ≥ 2r and r ≤ i ≤ d -r, that appears in the following theorem was proved, with a different technique, in [9, Theorem 1.7]. Theorem 3.5. For integers r, d and i such that d ≥ 2r, r ≤ i ≤ d -r, we have

Lemma 3 . 6 .

 36 Given r, d ∈ N, u ∈ S 1 , f ∈ S d and I ∈ Hilb r (P n ) such that u = 0 is a non-zero divisor for I, there exists a flat extension f ∈ Γ m+m ′ r of f such that u m+m ′ -d • f * = f * , and we have K d,0 r ⊂ E d,0 r . Proof. Let f ∈ S d with f * ∈ (I d ) ⊥ , I ∈ Hilb r (P n ).

  ) ⊥ ≃ (S d /I d ) * ≃ (R d /I ≤d ) * . a linear form f * ∈ (I d ) ⊥ corresponds to a linear form f * ∈ (R d /I ≤d ) * . As R d /I ≤d ֒→ R/I, the linear form f * ∈ (R d /I ≤d ) * can be extended (not necessary in a unique way) to a linear form φ ∈ (R/I) * .As the ideal I φ = ker H φ contains I and dim R/I = r, we deduce that H φ is of rank ≤ r. By restriction, H m,m φ is of rank ≤ r, where φ = φ |R ≤2m . The restriction of φ to R ≤d is f * .For any [f ] ∈ K d,0 r , there exists I ∈ Hilb r (P n ) such that f * ∈ (I d ) ⊥ . As I is a saturated ideal, it is always possible to find u ∈ S 1 such that u = 0 is nonzero divisor for I. By the previous construction, we can find a flat extension[ f ] ∈ Γ m+m ′ r of [f ], which this shows that [f ] ∈ E d,0 r and that K d,0 r ⊂ E d,0 r . Theorem 3.7. Let f ∈ S d .The following are equivalent:• There exists a zero-dimensional saturated ideal I defining ≤ r points counted with multiplicity such that I ⊂ (f ⊥ ); • f has a generalized decomposition of size ≤ r; • f has a flat extension of size ≤ r. In other words,[START_REF] Brachat | Symmetric Tensor Decomposition[END_REF] G d,0 r = K d,0 r = E d,0 r . and r G 0 (f ) = r sch 0 (f ) = r E 0 (f ).

Proof.

  By Lemma 3.1, we have G d,0 r ⊂ K d,0 r . By Lemma 3.6, we haveK d,0 r ⊂ E d,0 r . Let us prove that E d,0 r ⊂ G d,0 r . For any [f ] ∈ E d,0 r , there exists [ f ] ∈ Γ m,m r and u ∈ S 1 such that u m-r • f * = f * . By Theorem 3.5, [ f ] ∈ G 2m,0 r. Thus after some change of coordinates, f has an affine generalized decomposition of the form:f * = i=1,...,m 1 ζ i • φ i (∂). As f * = u m-r • f * , we have f * = u 2m-r • f * = i=1,...,m u 2m-r • (1 ζ i • φ i (∂)) = i=1,...,m

  is included in the inverse system spanned by1 ζ i • φ i (∂). Thus f has an affine decomposition f * = i=1,...,m 1 ζ i • φ ′ i (∂) of size ≤ r and [f ] ∈ G d,0 r . This shows that G d,0 r ⊂ K d,0 r ⊂ E d,0 r ⊂ G d,0r . and concludes the proof of the theorem.Corollary 3.8. G d r = K d r = E d r and r G (f ) = r sch (f ) = r E (f ). Corollary 3.9.For any homogeneous polynomial f we have that:

Remark 3 . 11 .

 311 At the beginning of the proof of Theorem 3.5, we showed that if d ≥ 2r and r ≤ i ≤ d -r, then G d,0 r = G d r , K d,0 r = K d r and E d,0 r = E d r .

  Tensor decomposition problem. The main problem we are interested in, is the problem of decomposition of a symmetric tensor into a sum of minimal size of indecomposable terms which are the powers of a linear forms: Definition 1.9. An element f ∈ S d has a decomposition of size r if there exist distinct non-zero elements k 1 , . . . , k r ∈ S 1 such that

	(2)

,-1) . Lemma 1.8. Suppose that I is a saturated ideal defining r simple points [ζ 1 ], . . . , [ζ r ] ∈ P n . Then (I d ) ⊥ is spanned by 1 ζ 1 , . . . , 1 ζr for d ≥ r. Proof. Obviously 1 ζ 1 , . . . , 1 ζr ⊂ (I d ) ⊥ . Moreover, as already observed in Remark 1.3, we have that dim(I d ) ⊥ = H S/I (d). Therefore, for d ≥ r, dim(I d ) ⊥ = r = dim 1 ζ 1 , . . . , 1 ζr and (I d ) ⊥ = 1 ζ 1 , . . . , 1 ζr . 1.2.

,...,ǫζ kn n )

Acknowledgement. We thank A. Bostan for the precise and detailed information he provided on combinatoric results related to the estimation of the catalecticant rank. We also thank J. Buczyński for many useful discussions and suggestions.

Definition 2.26. For any integers r and d, we define K d,0 r ⊂ P(S d ), as the set K d,0 r := {[f ] ∈ P(S d )| ∃ s ≤ r, ∃ I ∈ Hilb s (P n ), I d , f = 0}. In [START_REF] Buczyńska | Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes[END_REF], the r-cactus variety K d r is defined as the closure of K d,0 r . Definition 2.27 (Scheme length and border scheme length). The smallest r such that [f ] ∈ K d,0 r is called the scheme length (or cactus rank in [START_REF] Ranestad | On the rank of a symmetric form[END_REF]) of f and it is denoted r sch 0 (f ). The smallest r such that [f ] ∈ K d r is called the border scheme length of f and it is denoted r sch (f ).

We have used the same definition of "scheme length" of f used in [START_REF] Iarrobino | Power sums, Gorenstein algebras, and determinantal loci[END_REF][Definition 5.1, p. 135, Definition 5.66, p. 198], where it is denoted lsch(f ).

Example 2.28.

Thus, the scheme length of f is:

.

This is an example where the border rank and scheme length coincide but they differ from the rank (see Example 2.4). In the next example, we have a case where the scheme length is strictly smaller that the border rank.

Example 2.29. In the case of cubic polynomials, the scheme length of a generic form is smaller than its border rank for forms in 9 variables. The border rank of a generic cubic form in 9 variables is in fact 19 (this is Alexander and Hirschowitz Theorem [START_REF] Alexander | Generic hypersurface singularities[END_REF]), while the scheme length is smaller or equal than 18 (see [START_REF] Bernardi | On the cactus rank of cubic forms[END_REF]).

The generalized decomposition

The objective of this section is to relate the scheme length, generalized rank and flat extension rank. Lemma 3.1. Given two integers r and d, we have G d,0 r ⊂ K d,0 r . Proof. To prove this inclusion, we show that if g • f * has a generalized affine decomposition of size r of the form:

By a change of coordinates, we can assume that g = Id n+1 . Then the linear form: Λ = i=1,...,m

coincides with f * on R ≤d . By Proposition 1.6, as the dimension of the inverse system generated by Λ is s ≤ r, I = ker H Λ ⊂ R is a zero-dimensional ideal of


Norbert Weiss

Shahid Hameed

José M Fernández-Fernández

Katell Fablet

Maria Karmazinova

Cathy Poillot

Juliane Proft

Lina Chen

Isabelle Bidaud

Arnaud Monteil

Sylvaine Huc-Brandt

Lubica Lacinova

Philippe Lory

Dr Gerald W Zamponi
email: zamponi@ucalgary.ca

Dr Michel De Waard
email: dewaardm@ujf-grenoble.fr

A Ca(v)3.2/syntaxin-1A signaling complex controls T-type channel activity and low-threshold exocytosis

Keywords: calcium channel, T-type, Ca v 3.2 channel, SNARE, syntaxin-1A, SNAP-25, exocytosis, neuron, MPC cell

Background: T-type calcium channels play a unique role in low-threshold exocytosis. Results: Syntaxin-1A interacts with the carboxyterminus domain of Ca v 3.2 channels and modulates channel activity and low-threshold exocytosis. Conclusion: Low-threshold exocytosis relies to a syntaxin-1A/T-type calcium channel signaling complex.

INTRODUCTION

Ca 2+ entry through the high-voltageactivated (HVA) Ca v 2.2 (N-type) and Ca v 2.1 (P/Qtype) Ca 2+ channels into presynaptic nerve terminals supports a transient Ca 2+ microdomain that is essential for synaptic exocytosis (1)(2)(3). To ensure fast and efficient neurotransmitter release, the vesicle-docking / release machinery must be located near the source of Ca 2+ entry. In many cases, this close localization is achieved by direct interaction of SNARE proteins with the Ca 2+ channels (4)(5)(6). Indeed, syntaxin-1A/1B, SNAP-25 and synaptotagmin-1 specifically interact with Ca v 2.1 and Ca v 2.2 channels by binding to a synaptic protein interaction site (synprint) located within the intracellular II-III linker region of the channels (7)(8)(9)(10)(11), and disruption of Ca 2+ channel-SNARE coupling alters neurotransmitter release (6,12,13). A perhaps equally important consequence of SNARE protein-Ca 2+ channel interactions is the ability of SNARE proteins to modulate presynaptic Ca 2+ channel activity, thus fine tuning the amount of Ca 2+ that enters the synaptic terminal. Specifically, the binding of syntaxin-1A and SNAP-25 to Ca v 2.1 and Ca v 2.2 subunits shifts the voltage-dependence of inactivation toward more hyperpolarized membrane potentials to reduce channel availability (11,(14)(15)(16)(17)(18), and this effect is modulated by other members of the vesicle release complex (15,16,19,20).

In contrast with HVA channels, lowvoltage-activated (LVA) T-type Ca 2+ channels represent an important source of Ca 2+ entry near the resting membrane potential. It is established that T-type Ca 2+ currents mediate low-threshold burst discharges that occur during physiological neuronal rhythmogenesis as well as during pathological states such as epilepsy (21), and that they regulate various cellular functions such as neuronal development, dendritic integration, sensory transmission or pain signaling (for review see (22)). Furthermore, there is an increasing body of evidence that T-type channels can trigger fast and low-threshold exocytosis (23)(24)(25)(26)(27). In contrast with HVA Ca 2+ channels, all members of the Ttype channel family (i.e., Ca v 3.1, Ca v 3.2 and Ca v 3.3) lack the synprint site, and it is unclear as to how T-type channels are coupled to the exocytosis process at the molecular level.

Here, we reveal the existence of a syntaxin-1A/Ca v 3.2 T-type Ca 2+ channel complex in central neurons and show that this interaction involves different molecular determinants from those found in HVA channels. This interaction of syntaxin-1A with the Ca v 3.2 subunit potently modulates channel gating properties and appears essential for T-type channel-triggered lowthreshold exocytosis.

EXPERIMENTAL PROCEDURES

Plasmid cDNA constructs-The cDNAs constructs used in this study were human Ca v 3.1, Ca v 3.2 and Ca v 3.3 (28,29), mouse syntaxin-1A (Stx-1A) and SNAP-25 subcloned in pcDNA3 vector. The pcDNA3 construct encoding for the botulinium neurotoxin C1 (BoNT/C1) was generously provided by Dr. Robert G. Tsushima and Dr. Herbert Gaisano. The Stx-1A construct lacking the transmembrane domain (Stx1A TM , amino acids 1 to 265) in pCMV5 was kindly provided by Dr. Randy D. Blakely. The Stx1A"Open" in pMT2 was generated by mutating Leu 165 and Glu 166 to Ala as previously described (11).

Heterologous

Expression and Electrophysiology-Human embryonic kidney tsA-201 cells were grown in a Dulbecco's modified Eagle's culture medium containing 10% fetal bovine serum and 1% penicillin/streptomycin (all products were purchased from Invitrogen) and maintained under standard conditions at 37°C in a humidified atmosphere containing 5% CO 2 . Cells were transfected using the jetPEI™ transfection regent (Qbiogen, OH, USA) with either the Ca v 3.1, Ca v 3.2 or Ca v 3.3 channel, along with a green fluorescent protein (pEGFP, Clontech, CA, USA). Patch-clamp recordings were performed 48-72 hrs after transfection as previously described (30) Na-ATP, 10 HEPES (pH 7.4 with CsOH). 10 mM stock solution ErCl 3 (Sigma-Aldrich, St. Louis, MO, USA) was prepared daily in deionized water and diluted in the bath solution at a concentration of 30 M prior to the experiment. Only the cells in which series resistance was below 5 M were used for measurement of gating currents. Series resistance and capacitive transients were partly compensated by built-in circuits of the EPC 10 amplifier. Data were sampled at 10 kHz and filtered at 3 kHz. The linear component of the leak current and non-compensated capacitive transient was subtracted using the P/8 procedure. Total charge transferred during each pulse was evaluated by integrating the area below gating current trace at the beginning (Q on ) and after the end of each depolarizing pulse (Q off ). For capacitance measurements, MPC 9/3L-AH cells were transfected using Lipofectamine Plus (Invitrogen), as previously reported (6,31). The Ca v 3.2 channel was co-expressed with rat 2a and rabbit 2 subunits along with either the pEGFP, CD4-EGFP-Ca v 3.2 II-III or CD4-EGFP-Ca v 3.2 Cter construct, using a ratio of 1:1:1:0.6 respectively. Recordings were performed 48-72 hrs after transfection. The external solution contained (in millimolar): TEA-Cl 140, CaCl 2 5, HEPES 10, D-glucose 10 (pH 7.3; 300 mosmoles/l) (CaCl 2 was omitted and 1mM EGTA was added for studies in 0 Ca 2+ ). Patch pipettes were filled with a solution containing (in millimolar): 13 CsCl, 120 CH 3 CO 2 Cs, 2.5 MgCl 2 , 10 HEPES, 1 EGTA, 4 Na 2 ATP, 0.1 Na 3 GTP (pH 7.2; 295-300 mosmoles/l) and had a resistance of 2-4 M . Capacitance measurements were performed using an EPC-10 patch-clamp amplifier using the "sine + dc" software lock-in amplifier method implemented in PatchMaster software. MPC 9/3L-AH cells were held at -90 mV except during membrane depolarization. The sinusoid had amplitude of 25 mV and a frequency of 1 kHz. The stimulation protocol consisted in a single depolarization to -20 mV for 80 ms. Single-pulse current data were leak-subtracted by hyperpolarizing sweeps. Capacitance changes in response to ionomycin were recorded from MPC 9/3L-AH cells transfected with either pEGFP or CD4-EGFP-Ca v 3.2 Cter construct only. Ionomycin (1µM) was dissolved in the external solution indicated above and applied using a fast perfusion system (ALA VC 3 -8SP; ALA Scientific Instruments, Inc., NY). Analysis was performed using Igor Pro software version 6.0 (Wavemetrics).

Co-immunoprecipitation

experiments-Frozen rat brain tissue was homogenized in (10% w/v) of lysis buffer (in millimolar: 150 NaCl, 50 Tris, pH 7.5 and 1% NP-40 including a protease inhibitor cocktail (Roche)). The homogenate was centrifuged at ~16 000 g for 10 min at 4°C in a micro centrifuge and supernatant collected. For immunoprecepitation, 200 µL of aliquots of supernatant pre-cleared by addition of 40 µL of a 50% slurry (v/v) of rehydrated protein A-Sepharose beads (Amersham Pharmacia), followed by rotation at 4°C for 1 hr. Pre-cleared supernatants were then separately incubated with 5 µg of either anti-Ca v 3.1, anti-Ca v 3.2 or anti-Ca v 3.3 antibodies for 16 hrs at 4°C. To collect immuneprecipitated complex, the samples were further incubated for 2 hrs with 40 µL protein A-Sepharose beads (50% slurry). The beads with immunoprecipitated complex were washed four times with ice cold lysis buffer and resuspended in 40 µL of 2X SDS PAGE sample buffer. Heated at 70°C for 15 min and the soluble proteins were then resolved by SDS-PAGE and analyzed by Western blotting. Affinity-purified, rabbit polyclonal antibodies recognizing Ca v 3.1, Ca v 3.2 and Ca v 3.3 4 were purchased from Alomone Lab and mouse monoclonal syntaxin 1A antibody was from Chemicon International (Temecula, CA, USA). Co-immunoprecipitation experiments from tsA-201 cells were performed from cells expressing Stx1A-myc or SNAP-25 along with the CD4-EGFP-Ca v 3.2 Cter fusion protein were solubilized with 400 L of 50mM Tris/HCl pH 7.5, 0.05 mM EDTA and 10 mM Chaps Buffer supplemented with proteases inhibitor (1X, Roche) and lysed by sonication. Lysates were cleared by centrifugation at ~8000 g for 10 min. All steps were carried out on ice or at 4°C. Lysates were incubated for 3 hrs with 15 L biotinylated anti-Myc antibody (Santa Cruz) or for 16 hrs with anti-GFP antibody (1/1000, Abcam) and then for 45 min with streptavidine beads or protein G beads (Invitrogen) at 4°C and were washed twice with PBS (1X) Tween 0.1% and twice with PBS (1X). The beads were finally resuspended in PBS (1X). Samples were resolved on Tris-glycine SDSpolyacrylamide gels and transferred to nitrocellulose membranes using standard methods. Membranes were blocked in 10% (w/v) milk in PBS (0.1% (v/v) Tween-20) and then incubated with either anti-GFP or anti-SNAP-25 primary antibodies for 2 hrs at room temperature. After being washing once for 5 min, membranes were incubated with horseradish peroxidase conjugated protein A (Abcam). The washed membranes were developed using enhanced chemiluminescence.

nRT culture, immunostaining and confocal imaging-The Nucleus reticularis area was isolated from lateral thalamus of P0-P1 rat pups, cut into small pieces and then digested in containing papain (Worthington, LS003126) culture media. After digestion, the tissue was washed and triturated for neuron dissociation. nRT neurons were seeded at low density onto coverslips pretreated with poly-D-lysine (Sigma, P7280) followed by Laminin (Sigma, L2020) in 24-well plate. Culture medium consisted of BME (Invitrogen, 21010) supplemented with 1% B-27 supplement (Invitrogen, 17504-044), 2mM L-glutamine (Invitrogen, 25030-049), 1mM sodium pyruvate (Invitrogen, 11360-070), 10 mM HEPES(Sigma, H7523), 0.6% D-glucose (Sigma, G7528), 1% pen strep (Invitrogen, 15140), and 5% fetal bovine serum (Invitrogen, 26140). The next day after plating, the existing media was replaced by fresh culture media, after that every 4-5 days half of the media was removed and replaced by fresh culture media. For immunostaining, cells were fixed with 4% paraformaldehyde in PBS for 20 min at room temperature, washed and permeabilized in PBST (PBS containing 0.02% Triton X-100) for 3 min, and blocked with blocking buffer (3% BSA in PBS) for 30 min. Cells were then incubated overnight at 4°C with primary polyclonal rabbit anti-Ca v 3.2 (Alomone) and mouse monoclonal anti-syntaxin-1A (Sigma) antibodies diluted at 1:400 and 1:500, respectively, in blocking buffer. Cells were then washed, incubated 1h at room temperature with anti-rabbit Alexa488-conjugated and anti-mouse Alexa555-conjugated secondary antibodies, washed and mounted on glass slide. Confocal images were acquired with a Zeiss LSM510 Meta microscope and analysed using ImageJ software. The intensity correlation analysis was performed using the ImageJ plugin previously described (32).

Statistics-Data values are presented as mean ± S.E.M. for n recorded cells. Statistical significance was determined using Student's t test: * p < 0.05; ** p < 0.01; *** p < 0.001; NS, statistically not different.

RESULTS

Syntaxin-1A associates with Ca v 3.2 channels in central neurons -The existence of a relationship between T-type Ca 2+ channels and neurotransmitter release (23)(24)(25)(26) prompted us to examine the possible colocalization of these channels with SNARE proteins in central neurons. Isolated rat reticular thalamic neurons (nRT) that are known to express predominantly the Ca v 3.2 isoform of T-type channels (33) were immunolabeled, and T-type channel and syntaxin-1A distribution analyzed by confocal microscopy. The anti-Ca v 3.2 antibody labeling (green) revealed a somatodendritic pattern consistent with previous electrophysiological studies (34)(35)(36). Interestingly, intense syntaxin-1A labeling was also similarly observed along the somatodendritic axis (red) and visual inspection of the overlaid images and colocalized pixels suggests colocalization of the proteins (Fig. 1A). We used the intensity correlation analysis (ICA) method to test for a staining relation between Ca v 3.2 channel and syntaxin-1A as previously described (32). The ICA plots of Ca v 3.2 and syntaxin-1A performed both on the soma and on dendrite regions were strongly skewed toward positive values (Fig. 1B), consistent with a highly dependent staining pattern. Furthermore, the calculated intensity correlation quotient values (ICQ) were positive and highly significant (+0.30 ± 0.01 and +0.31 ± 0.01 for the soma and dendrite respectively (n = 12; p < 0.001). These data thus strongly suggest that a pool of syntaxin-1A and Ca v 3.2 channels are present at the same subcellular level. We then investigated by coimmunoprecipitations whether T-type calcium channels and syntaxin-1A associate with each other at the protein level. As shown in Figure 1B, specific anti-Ca v 3.1, Ca v 3.2 and Ca v 3.3 antibodies precipitated syntaxin-1A from rat brain homogenate, thus suggesting the existence of Ttype Ca 2+ channel/syntaxin-1A complexes (Fig. 1C).

Syntaxin-1A modulates Ca v 3.2 channel activity -To investigate the impact of the Ca v 3.2/syntaxin-1A interaction on channel function, channel properties were analyzed in the absence and the presence of syntaxin 1A using whole-cell patch-clamp. The co-expression of syntaxin-1A with Ca v 3.2 channel in tsA-201 cells reveals a potent syntaxin-1A regulation of the voltage-dependence of T-type channel inactivation (Fig. 2A-C and Table S1). In syntaxin-1A expressing cells, the half-inactivation potential of Ca v 3.2 channels was shifted to more hyperpolarized potentials by -16.3 ± 1.2 mV (p < 0.001) (to -86.9 ± 1.2 mV, n = 28) compared to control cells (-70.6 ± 0.5 mV, n = 27). The syntaxin-1A-mediated modulation of Ca v 3.2 inactivation was abolished upon cleavage of syntaxin-1A by co-expression of the botulinium neurotoxin C1 (BoNT/C1) (Fig. 2C and Table S1), indicating that the membrane insertion domain of syntaxin-1A was necessary for the observed functional effects. Consistent with this observation, co-expression of a syntaxin-1A construct lacking the transmembrane domain was unable to modulate Ca v 3.2 channel inactivation (Fig. 2C and Table S1). Similar results were obtained with Ca v 3.1 and Ca v 3.3 channels, indicating that syntaxin-1A regulation is conserved across the entire T-type channel family (Fig. S1).

It is well established that syntaxin-1A exists in both "open" and "closed" conformations (37), and that syntaxin-1A state is an important determinant of N-type channel modulation (11). Similar to previous observations in N-type channels, co-expression of a mutant syntaxin-1A construct that is locked permanently in the "open" conformation failed to induce a hyperpolarizing shift of the steady-state in half-inactivation potential (Fig. 2C), indicating that syntaxin-1A mediated modulation of Ca v 3.2 channel is syntaxin state-dependent.

Besides regulating the voltage-dependence of Ca v 3.2 channel inactivation, syntaxin-1A also produced moderate effects on other gating properties of the channel, including a -5.7 mV hyperpolarizing shift of the activation curve (from -38.6 ± 0.6 mV (n = 31) to -44.2 ± 0.8 mV (n = 18), p < 0.001) (Fig. 2D and Table S1), a 1.6-fold slowing of the time constant of inactivation kinetics (from 13.1 ± 0.7 ms (n = 31) to 21.0 ± 1.3 ms (n = 18) at -20 mV, p < 0.001) (Table S1), and a 1.8-fold slowing of the time constant for recovery from fast inactivation (from 489.4 ± 25.9 ms (n = 25) to 876.3 ± 60.7 ms (n = 14), p < 0.001) (Table S1). In contrast, the voltage-dependence and kinetics of intramembrane charge movements of Ca v 3.2 channel were found unaffected upon coexpression of syntaxin-1A (Fig. S2). Unlike syntaxin-1A, co-expression of other synaptic proteins such as SNAP-25, synaptobrevin-1 and 2, CSP and Rim-1 produced only minor effects on Ca v 3.2 channel activity (Table S1). Altogether, our data indicate that syntaxin-1A is a robust and selective modulator of T-type Ca 2+ channel inactivation.

Syntaxin-1A interacts within the carboxyterminal domain of the Ca v 3.2 subunit -Previous studies have shown that functional interaction of N-and P/Q-type Ca 2+ channels can be disrupted by a peptide containing the synprint site (12,13). In order to investigate the Ca v 3.2 molecular determinants of syntaxin-1A regulation, we examined the capability of the different intracellular regions of the Ca v 3.2 subunit to prevent syntaxin-1A regulation. T-type Ca v 3.2 channels were coexpressed in tsA-201 cells along with syntaxin-1A +/-a construct in which the major intracellular regions of the channels were fused to a CD4 epitope as well as the green fluorescent protein (EGFP) (Fig. 3A). Barium currents were recorded at -20 mV from a holding potential of -85 mV (where 56% of the channels are tonically inactivated in the presence of syntaxin-1A versus 11% in the absence of syntaxin-1A) before and after a 5 sec hyperpolarizing prepulse to -130 mV. A 5 sec hyperpolarizing prepulse to -130 mV produced a 2.1-fold increase in current amplitude in cells expressing syntaxin-1A (versus 1.1-fold increase in the absence of syntaxin-1A) due to the recovery of channels from inactivation (Fig. 3B). This prepulse-induced current facilitation was found significantly decreased upon co-expression of the CD4-EGFP-Ca v 3.2 Cter fusion protein (from 2.1 ± 0.2-fold (n = 20) to 1.4 ± 0.1 (n = 17), p < 0.001), whereas expression of the CD4-EGFP-Ca v 3.2 Cter construct by itself did not affect Ba 2+ currents in the absence of syntaxin-1A (indicating that the CD4-EGFP-Ca v 3.2 Cter construct by itself does not alter the voltage-dependence of inactivation of Ca v 3.2 channels) (Fig. 3B). In contrast, expression of the other intracellular regions of the channel failed to prevent syntaxin-1A regulation (Fig. 3B), suggesting a potential interaction of syntaxin-1A within the carboxy-terminus of the channel. Consistent with a possible specific binding of syntaxin-1A within the C-terminal domain of Ca v 3.2, co-expression of syntaxin-1A in COS cells promoted translocation of a EGFP-Ca v 3.2 Cter fusion protein to the plasma membrane whereas the EGFP-Ca v 3.2 II-III linker fusion protein remained diffusely expressed inside the cell (Fig. 3C). The trafficking study was performed on COS cells because they present the advantage to have a smaller nucleus than tsA-201 cells and more diffuse cytoplasm, allowing a better analysis of protein trafficking and plasma membrane staining. Finally, co-immunoprecipitation experiments from tsA-201 cells using the CD4-EGFP-Ca v 3.2 Cter fusion protein revealed that syntaxin-1A specifically interacts with the carboxy-terminus domain of the Ca v 3.2 subunit (Fig. 3D), which is in stark contrast with the reported syntaxin-1A interaction site on HVA channels (II-III loop). Altogether, these findings demonstrate that syntaxin-1A modulates Ca v 3.2 channel activity by interacting within the C-terminal domain of the channel and that co-expression of the putative interaction site of the channel competitively inhibits syntaxin-1A-induced channel modulation.

Like syntaxin-1A, SNAP-25 was also found to associate with the Ca v 3.2 C-terminus (Fig. S3). While SNAP-25 by itself produced only slight effects on T-type channel activity, its coexpression reversed the effects of syntaxin-1A on channel inactivation properties (Fig. S3 and Table S1), in analogy with previous observations with Ntype channels (15,16).

Ca v 3.2-SNARE complexes contribute to low-threshold exocytosis

-Besides conferring a regulation of channel availability, the interaction between syntaxin-1A and T-type channels may also serve as a means for optimizing the coupling between T-type channel-mediated Ca 2+ entry and secretion. To test for this possibility, Ca v 3.2 channels were over-expressed in MPC 9/3L-AH cells, a chromaffin cell line already fully endowed with the vesicle release machinery, but devoid of voltage-gated Ca 2+ channels (6). Voltagedependent exocytosis was evaluated by monitoring membrane capacitance changes (indicative of vesicle fusion) in response to a 100 ms depolarizing step to -20 mV from a holding potential of -90 mV. Voltage-dependent activation of over-expressed Ca v 3.2 channels produced a robust increase in exocytosis (estimated by normalizing capacitance changes by Ca 2+ charge density) which was reduced by 96% (p < 0.001) when SNARE proteins were uncoupled from the channel by co-expression of the CD4-EGFP-Ca v 3.2 Cter fusion protein (Fig. 4C-D), even though Ca 2+ entry per se was not affected (1.9 ± 0.7 pC/pF, (n = 10) versus 1.8 ± 0.4 pC/pF, (n = 15)) (Fig. 4A-B). In contrast, co-expression of the CD4-EGFP-Ca v 3.2 II-III linker fusion protein, which does not interact with syntaxin-1A, had no effect on Ca v 3.2-induced voltage-dependent exocytosis. The observation that Ca 2+ entry was not affected upon co-expression of the CD4-EGFP-Ca v 3.2 Cter fusion protein is consistent with the notion that syntaxin-1A modulation is abolished when Ca v 3.2 channels are functionally coupled to a full SNARE complex.

To confirm that the decrease in exocytosis observed upon expression of the CD4-EGFP-Ca v 3.2 Cter fusion protein is due to the uncoupling of 7 Ca v 3.2 channels from endogenous SNARE proteins and not to a direct distortion of the vesicle release machinery, exocytosis was evaluated in EGFP-and CD4-EGFP-Ca v 3.2 Cter -expressing MPC 9/3L-AH cells (in the absence of Ca v 3.2 channel) in response to a Ca 2+ ionophore. Exposition of EGFP-expressing cells to ionomycin (1 µM) elicited an increase in membrane capacitance that was significantly reduced by 94% (p < 0.001) after removal of extracellular Ca 2+ (273 ± 95 fF, (n = 13) versus 17 ± 11 fF, (n = 9)) (Fig. 5A-B). However, neither the magnitude (273 ± 95 fF, (n = 13) versus 288 ± 81 fF, (n = 11), p = 0.37), nor the kinetic (tau) of the exocytosis response (9.0 ± 2.1 sec versus 8.9 ± 1.9 sec, p = 0.98) was affected by overexpression of the CD4-GFP-Ca v 3.2 Cter construct (Fig. 5A-C). These results indicate that the negative impact of the CD4-EGFP-Ca v 3.2 Cter fusion protein on the voltage-dependent exocytosis from MPC 9/3L-AH cells expressing Ca v 3.2 channels is not due to a direct distortion of the exocytotic machinery by the fusion protein but rather to the uncoupling of the channel from endogenous SNARE proteins.

Altogether, these findings indicate that Ca v 3.2 channels have the propensity to support low-threshold exocytosis, and that this process requires a close proximity between the vesicle release machinery and the channel.

DISCUSSION

In the present study, we provide compelling evidence for the existence of a syntaxin-1A/Ca v 3.2 T-type Ca 2+ channel signaling complex. We demonstrate that this interaction not only potently modulates Ca v 3.2 channel gating but also regulates T-type channel mediated exocytosis.

It was previously shown that Ca v 2.1 and Ca v 2.2 Ca 2+ channels colocalize with syntaxin-1A at presynaptic nerve terminals (38)(39)(40) and that these channels can be isolated as a complex with SNARE proteins (41)(42)(43). We show that similarly to Ca v 2.1 and Ca v 2.2 channels, Ca v 3.2 channels also colocalize with syntaxin-1A along the somatodendritic axis of nRT neurons, and that these two proteins can be co-immunoprecipitated, thus suggesting the existence of a native syntaxin-1A/Ca v 3.2 channel signaling complex. Moreover, in vitro binding experiments indicate that syntaxin-1A interacts within the carboxy-terminal domain of the Ca v 3.2 subunit, which indicates that T-type channels use different structural determinants compared with the synprint site utilized by members of the HVA channel family. At this point it is unclear whether distinct syntaxin-1A structural determinants are involved in the interactions with Ca v 3 and Ca v 2 channels.

Electrophysiological analyses of Ca v 3.2 channels in tsA-201 cells revealed that syntaxin-1A potently decreases channel availability by shifting the steady-state inactivation toward more hyperpolarized potentials. This is qualitatively similar to what has been reported for N-and P/Qtype Ca 2+ channels. Our observation that coexpression of the carboxy-terminal region of the Ca v 3.2 subunit prevents syntaxin-1A-mediated modulation of Ca v 3.2 channel strongly suggest that syntaxin-1A modulates channel activity by virtue of its direct binding to the carboxy-terminal of the Ca v 3.2 subunit, rather than the mere presence of a syntaxin-1A molecule in the cell. The notion that syntaxin-1A-mediated modulation of Ca v 3.2 channel inactivation is abolished by botulinium neurotoxin C1 or upon deletion of the syntaxin-1A transmembrane domain indicates that membrane anchoring of syntaxin-1A is critical for channel modulation, and is analogous with previous observations on HVA channels (44). The inability of the "open" form of syntaxin-1A to alter T-type channel gating also parallels previous observations in Ca v 2.2 (11). Given that syntaxin-1A undergoes a conformational switch from a "closed" state to an "open" during the vesicle release cycle (37,45,46), this suggests the possibility that syntaxin-1A may be able to dynamically regulate Ca v 3.2 channel availability during various stages of exocytosis. The observation that SNAP-25 coexpression could reverse the syntaxin-1A mediated effect on channel gating is reminiscent of findings with Ca v 2.2 (but not Ca v 2.1) channels (14)(15)(16). At this point, we do not know whether this is due to a direct competition between SNAP-25 and syntaxin-1A for the same binding site within the C-terminus region of the channel, or whether both proteins can bind to the channel simultaneously. Irrespective of these structural information, the syntaxin-1A mediated alteration in the voltage-dependence of inactivation and the ability of SNAP-25 to regulate this process may represent an important regulatory pathway for T-type Ca 2+ channels. This is further underscored when considering that these channels are ~50% inactivated near neuronal resting membrane potentials such that even small changes in the voltage dependence of inactivation can bring about dramatic changes in Ca 2+ entry (47).

Besides modulating T-type Ca 2+ channel gating, the physical association between Ca v 3.2 channel and SNARE proteins may contribute significantly to optimize Ca v 3.2-mediated exocytosis. Consistent with previous studies showing that Ca v 2.1, Ca v 2.2 or Ca v 3.1 channels can trigger voltage-dependent exocytosis when heterologously expressed in MPC-9/3L cells (6,31,48), we show that expression of Ca v 3.2 channel in these cells induced robust voltagedependent exocytosis that was drastically reduced when SNARE proteins were competitively uncoupled from the channel by co-expression of the carboxy-terminal domain of the Ca v 3.2 subunit. This result is consistent with previous studies showing that disruption of N-and P/Q-type Ca 2+ channels-SNAREs interaction by a peptide containing the synprint site prevents neurotransmitter release (12,13). It was proposed that uncoupling of Ca 2+ channels from SNARE proteins may not only lead to a spatial dissociation of vesicles and channels, but also to a loss of feedback regulation that collectively alter exocytosis efficiency (49,50). In this context, our data fit with the idea that T-type Ca 2+ channels support low-threshold exocytosis by virtue of their close association with the SNARE protein machinery. Such tight coupling may be particularly important for T-type channels due to their small unitary conductance and low open probability.

In conclusion, although HVA and T-type channels utilize completely different channel structural determinants for their interactions with the SNARE machinery, they are functionally regulated by syntaxin-1A in a strikingly similar manner. This may perhaps underscore the fundamental evolutionary importance for providing tight control over (and localizing the exocytosis machinery close to the source of) Ca 2+ entry. Overall, our findings reveal unrecognized new insights into the regulation of T-type Ca 2+ channels by SNARE proteins and provide the first molecular mechanism by which these channels contribute to low-threshold exocytosis.

FOOTNOTES

in 3 the

 3 whole-cell configuration of the patch-clamp technique at room temperature (22-24°C) in a bathing medium containing (in millimolar): 5 BaCl 2 , 5 KCl, 1 MgCl 2 , 128 NaCl, 10 TEA-Cl, 10 D-glucose, 10 HEPES (pH 7.2 with NaOH). Patch pipettes were filled with a solution containing (in millimolar): 110 CsCl, 3 Mg-ATP, 0.5 Na-GTP, 2.5 MgCl 2 , 5 D-glucose, 10 EGTA, 10 HEPES (pH 7.4 with CsOH), and had a resistance of 2-4 M . Whole-cell patch-clamp recording were performed using an Axopatch 200B amplifier (Axon Instruments, Union City, CA). All traces were corrected on-line for leak currents, digitized at 10 KHz and filtered at 2 KHz. Intramembrane charge movement measurements were recorded using an HEKA-10 patch clamp amplifier (HEKA Electronic, Lambrecht, Germany). The extracellular solution contained (in millimolar): 105 CsCl, 10 HEPES, 10 D-glucose, 10 TEA-Cl, 2 CaCl 2 , 1 MgCl 2 (pH 7.4 with CsOH). The intracellular solution contained (in millimolar): 130 CH 3 SO 3 Cs, 10 EGTA, 5 MgCl 2 , 10 TEA-Cl, 5

FIGURE 1 .

 1 FIGURE 1. Ca v 3.2 channel interacts with syntaxin-1A in central neurons. A, Confocal images of nRT neurons permeabilized and stained for Ca v 3.2 (green) and syntaxin-1A (red). Overlaid images and colocalized pixels (in white) are shown. B, Intensity correlation analysis (ICA) plots of Ca v 3.2 and syntaxin-1A staining intensities against their respective (A-a)(B-b) values performed from the ROIs indicated in the overlaid image in (A). IQC = +0.31 and +0.38 for soma and neurite regions respectively. C, Co-immunoprecipitation of syntaxin-1A from rat brain homogenate with specific anti-Ca v 3.1, anti-Ca v 3.2 and anti-Ca v 3.3 antibodies.

FIGURE 2 .

 2 FIGURE 2. Syntaxin-1A modulates Ca v 3.2 channel inactivation. A, Representative Ba 2+ current traces recorded from a Ca v 3.2 (top panel) and Ca v 3.2/Stx-1A-expressing cell (bottom panel) in response to 150 ms depolarizing steps to -20 mV from a holding potential varied from -120 mV to -50 mV (a to f). B, Corresponding mean normalized steady-state inactivation curves for Ca v 3.2 (filled circles) and Ca v 3.2/Stx-1A-expressing cells (open circles). C, Plot of the mean shift values in the half-inactivation potential of Ca v 3.2 channel coexpressed with the different protein combinations indicated in the figure. D, Mean normalized activation curve for Ca v 3.2 (filled circles) and Ca v 3.2/Stx1A-expressing cells (open circles). Inset indicates the shift values in the half-activation potential produced upon coexpression of Stx1A or Stx1A TM . Stx-1A, syntaxin-1A; BoNT/C, botulinium neurotoxin C1.

FIGURE 3 .

 3 FIGURE 3. Syntaxin-1A interacts within the carboxy-terminal domain of Ca v 3.2. A, Schematic representation of the different constructs of intracellular regions of Ca v 3.2 used. B, Whole-cell Ba 2+ currents (top panels) recorded in response to a 150 ms depolarizing step to -20 mV from a holding potential of -85 mV before (P1) and after (P2) a 5 sec hyperpolarizing pulse to -130 mV in a Ca v 3.2 (left panel), Ca v 3.2 / Stx-1A (middle panel) and Ca v 3.2 / Stx-1A / CD4-Ca v 3.2 Cter -expressing cell (right panel) and the corresponding mean plot of the current facilitation (I P2 / I P1 ) (bottom panel). Note that the hyperpolarizing pulse produces a strong current facilitation in the presence of Stx-1A which is competitively and specifically abolished upon co-expression of the CD4-Ca v 3.2 Cter construct. C, Confocal images of living COS cells showing the translocation of the EGFP-Ca v 3.2 Cter construct (green) to the plasma membrane mediated by Stx-1A. Plasma membrane was stained with rhodamine labeled concanavalin A (ConA-Rhod, red). Overlaid images and pixel intensity profiles of crossed sections indicated by the white line are shown. Note that Stx-1A does not translocate EGFP-Ca v 3.2 II-III linker fusion protein. D, Co-immunoprecipitation of the CD4-EGFP-Ca v 3.2 Cter fusion protein from tsA-201 cells cotransfected with Stx-1A-Myc. The upper panel shows the immunoblot of CD4-EGFP-Ca v 3.2 Cter fusion protein in the absence (-) and presence (+) of Stx-1A-Myc using an anti-GFP antibody. *Possible degradation of the CD4-EGFP-Ca v 3.2 Cter fusion protein. The lower panel shows the results of the coimmunoprecipitation of the CD4-EGFP-Ca v 3.2 Cter fusion protein with Stx-1A-Myc using an anti-Myc antibody. In the absence of Stx-1A-Myc the antibody alone is not able to immunoprecipitate the CD4-EGFP-Ca v 3.2 Cter fusion protein.

FIGURE 4 .

 4 FIGURE 4. Ca v 3.2 channel induces voltage-dependent exocytosis. A, Representative Ca 2+ current traces recoded from MPC 9/3L-AH cells expressing Ca v 3.2 channel alone (top panel), and in combination with CD4-Ca v 3.2 Cter (middle panel) or CD4-Ca v 3.2 II-II linker (bottom panel) in response to a 100 ms long depolarizing step to -20 mV from a holding potential of -90 mV. These data show that T-type currents are functional in the presence of CD4-Ca v 3.2 Cter . B, Corresponding mean Ca 2+ influx normalized by the whole-cell capacitance [Q Ca density (pC/pF)] elicited by the 100 ms long depolarizing step to -20 mV. C, Capacitance traces plotted as a function of time from the same cells shown in (A). D, Corresponding mean exocytosis [ Cm (fF)] normalized as a function of Ca 2+ entry [Q Ca density (pC/pF)].

FIGURE 5 .

 5 FIGURE 5. Expression of the CD4-EGFP-Ca v 3.2 Cter fusion protein does not prevent exocytosis induced by intracellular calcium elevation. A, Representative capacitance traces plotted as a function of time recorded from MPC 9/3L-AH cells expressing EGFP (black trace) or CD4-EGFP-Ca v 3.2 Cter (light grey) in response to the superfusion of 1µM ionomycin (indicated by the arrow). A representative capacitance trace recorded from an EGFP-expressing cell in the absence of extracellular Ca 2+ (EGFP -0 Ca 2+ ) is also shown (dark grey). B, Corresponding mean exocytosis values [ Cm (fF)] for EGFP (with (black) and without (dark grey) extracellular Ca 2+ ) and CD4-EGFP-Ca v 3.2 Cter -expressing cells (light grey). C, Corresponding time constant values (tau) of capacitance change obtained by fitting the rising phase of the traces by a single exponential function.

FIGURES

  FIGURES

Acknowledgments -We are grateful to Drs. Robert G. Tsushima (York University, Toronto, Canada) and Herbert Gaisano (University of Toronto, Toronto, Canada) for providing the BoNT/C1, and to Dr. Randy D. Blakeley (Vanderbilt University, Nashville, USA) for the Stx1A TM constructs. We thank Dr. Amy Harkins (Saint Louis University, Missouri, USA) for making available the MPC 9/3L-AH cell line and Dr. Cristina Plata (Universitat Pompeu Fabra, Barcelona, Spain) for technical assistance with MPC cells transfection. We thank Drs. Mike Seagar and Christian Lévêque (Inserm U641, Marseille, France) for antibodies. We are grateful to Dr. Emilio Carbone (University of Torino, Torino, Italy) for critical discussions.

NW was supported by fellowships from Alberta Heritage Foundation for Medical Research and from Hotchkiss Brain Institute. JMFF was funded by Spanish Ministry of Science and Innovation, Fondos Europeos de Desarrollo Regional (FEDER) Funds, and Plan E (Grant SAF2009-13182-C03-02), Fondo de Investigación Sanitaria (Red HERACLES RD06/0009) and Generalitat de Catalunya (Grant 2009SGR1369). LL was supported by VEGA 2/0195/10 grant. PL was supported by ANR-2006-Neuro35 and ANR-09-MNPS-035 grants. GWZ is funded by the Canadian Institutes of Health Research, is a Canada Research Chair and Scientist of the Alberta Heritage Foundation for Medical Research.