On Apolarity and Generic Canonical Forms

Journal of Algebra 213, 167᎐194 Ž1999. Article ID jabr.1995.6649, available online at http:rrwww.idealibrary.com on

On Apolarity and Generic Canonical Forms Richard Ehrenborg* Department of Mathematics, White Hall, Cornell Uni¨ ersity, Ithaca, New York 14853-7901 Communicated by D. A. Buchsbaum Received December 10, 1995

The notion of S-algebra is introduced. The theory of apolarity and generic canonical forms for polynomials is generalized to S-algebras over the complex field ⺓. We apply this theory to the problem of finding the essential rank of general, symmetric, and skew-symmetric tensors. Upper bounds for the essential ranks are found by different combinatorial coverings. 䊚 1999 Academic Press

1. INTRODUCTION The theory of apolarity was first developed by Clebsch, Lasker, Richmond, Sylvester, and Wakeford w10, 17, 20x. They were first interested in studying homogeneous polynomials of degree p and in q variables, and in expressing them as sums of pth powers of linear terms. The problem is to minimize the number of pth powers which are required in such a sum. For instance, a result due to Sylvester is that a generic homogeneous polynomial in two variables of degree 2 n y 1 may be written as a sum with n terms of Ž2 n y 1.st powers. This is only true for a generic polynomial; for instance x 2 y cannot be written as the sum of two cubes. They then focused on the more general problem of finding canonical ways of expressing a generic homogeneous polynomial. For example, it was

* The research for this paper was done at MIT, and the paper was written when the author was a joint postdoctoral fellow between Centre de Recherches Mathematiques at Universite ´ ´ de Montreal ´ and LACIM at Universite´ du Quebec ´ `a Montreal. ´ 167 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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RICHARD EHRENBORG

discovered that a generic quaternary cubic Žfour variables and degree 3. may be written in the form h1 h 2 h 3 q h 4 h 5 h 6 , where the h i ’s are linear terms. In w3x the main theorems on apolarity and generic canonical forms are presented for homogeneous polynomials, and numerous examples are included. In the present work we extend the theory of apolarity and generic canonical forms to a more general setting. We introduce the notion of S-algebra, which is a vector space A together with a set S of multilinear forms on the space A. This idea generalizes the notion of an algebra. We then consider the space of all functions, A x 1 , . . . , x d 4 , that may be constructed on the space A to itself by the multilinear forms in the set S . These functions behave like polynomials. We consider an important class of derivations on the space A x 1 , . . . , x d 4 , namely the polarizations. The polarizations satisfy many properties, including some chain rules which play a crucial role in the proof of our main theorem. Recall that symmetric tensors SymŽW . correspond to polynomials in n variables, where n is the dimension of W. The classical notion of apolarity can then be formulated as follows. Given two symmetric tensors f g Sym p ŽW *. s Sym p ŽW .* and g g Sym r ŽW ., where r F p, we say that g is apolar to f if for all h g Sym py r ŽW . we have that ² f < g ⭈ h: s 0. We restate this definition by considering the linear map ␾ : Sym py r ŽW . ª Sym p ŽW . defined by ␾ Ž h. s g ⭈ h. Now we say that the linear map ␾ is apolar to f g Sym p ŽW *. if for all h g Sym pyr ŽW . we have that ² f < ␾ Ž h.: s 0. It is in this setting that we are able to generalize the idea of apolarity; see Definition 4.4. In Section 5 we state the main theorem on generic canonical forms and apolarity. In short, the problem of determining whether a form p g A x 1 , . . . , x d 4 in an S-algebra A is canonical may be solved by finding certain elements in the vector space and showing that a certain linear equation system does not have a non-trivial solution. The concepts of apolarity and polarizations are important in finding this equation system. We apply the above theory to the study of the rank of general tensors, symmetric tensors, and skew-symmetric tensors. In Section 6 we consider general tensors, which correspond to multi-dimensional matrices. The problem of finding upper bounds for the essential rank of general tensors has a combinatorial interpretation. Namely, an upper bound for the number of rooks required to cover a multidimensional chess board is also an upper bound for the essential rank of tensors. Similar combinatorial bounds are found in Section 7 for symmetric and skew-symmetric tensors. With the help of Steiner triple systems we find explicit bounds for the essential rank of symmetric and skew-symmetric tensors of degree 3. In the concluding remarks we suggest a conjecture which gives a duality between symmetric and skew-symmetric tensors.

APOLARITY AND GENERIC CANONICAL FORMS

169

2. POLYNOMIAL FUNCTIONS DEFINITION 2.1. An S-algebra A is a vector space together with a set S of multilinear forms on A. That is, for each M g S there exists a positive integer k Ž M . such that M :^A =` ⭈⭈⭈ = A _ª A, is a multilinear form.

kŽM .

When there is no risk of confusion we will write k instead of k Ž M .. An example of an S-algebra is any associative algebra A. In this case S consists of a single bilinear form, which is the product on the algebra. But in the general instance of S-algebra we do not assume any relations among the multilinear forms in the set S . In this article we will consider S-algebras over the complex field ⺓. We assume that A carries a topology such that the induced topology on every finite dimensional linear subspace is Euclidean. Moreover, we also assume that S is a set of continuous multilinear forms on the linear space A. We define A x 1 , . . . , x d 4 to be the set of all functions in the variables x 1 , . . . , x d that we may construct by using the multilinear forms in the set S . More formally DEFINITION 2.2. Let A x 1 , . . . , x d 4 be the smallest set that contains the vector space A and the variables x 1 , . . . , x d , and which is closed under 1. linear combinations: if p, q g A x 1 , . . . , x d 4 and ␣ , ␤ g ⺓ then ␣ ⭈ p q ␤ ⭈ q g A x 1 , . . . , x d 4 , 2. compositions with the multilinear forms in the set S : if M g S, k s k Ž M . and p1 , . . . , pk g A x 1 , . . . , x d 4 then M Ž p1 , . . . , pk . g A x 1 , . . . , x d 4 . The elements of A x 1 , . . . , x d 4 are called polynomials in the d variables x1, . . . , x d . Observe that an element of A x 1 , . . . , x d 4 may be constructed from A and the variables x 1 , . . . , x d in a finite number of steps by the two rules in Definition 2.2. Hence we can prove statements about elements in A x 1 , . . . , x d 4 by induction. In the language of universal algebra the set A x 1 , . . . , x d 4 is the polynomial clone of d-ary operations closed under the functions in S , addition, and multiplication with scalars in ⺓; see w15x. LEMMA 2.3. There is a unique e¨ aluation map, denoted by eval, from A x 1 , . . . , x d 4 = A d to A such that for Ž a1 , . . . , a d . g A d 1. evalŽ x i ; a1 , . . . , a d . s a i , 2. evalŽ a; a1 , . . . , a d . s a for a g A,

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RICHARD EHRENBORG

3. evalŽ ␣ ⭈ p q ␤ ⭈ q; a1 , . . . , a d . s ␣ ⭈ evalŽ p; a1 , . . . , a d . q ␤ ⭈ evalŽ q; a1 , . . . , a d . for p, q g A x 1 , . . . , x d 4 and ␣ , ␤ g ⺓, 4. evalŽ M Ž p1 , . . . , pk .; a1 , . . . , a d . s M ŽevalŽ p1; a1 , . . . , a d ., . . . , evalŽ pk ; a1 , . . . , a d .. for M g S , k s k Ž M ., and p1 , . . . , pk g A x 1 , . . . , x d 4 . To avoid confusion we will sometimes write evalŽ p; x 1 ¤ a1 , . . . , x d ¤ . a d instead of the shorter Žbut still correct. evalŽ p; a1 , . . . , a d .. We may say that two polynomials p and q in A x 1 , . . . , x d 4 are equivalent if for all a1 , . . . , a d g A we have that eval Ž p; a1 , . . . , a d . s eval Ž q; a1 , . . . , a d . . That is, p and q behave in the same way as functions, but they may consist of different expressions.

3. POLARIZATIONS In this section we introduce an important class of linear maps on the linear space A x 1 , . . . , x d 4 . These maps generalize the concept of derivations on an algebra. This development will be done very much in the spirit of the theory of supersymmetric algebra; see, for instance, w7x. DEFINITION 3.1. A polarization Dt, x i is a linear map from the polynomials A x 1 , . . . , x d 4 to the polynomials A t, x 1 , . . . , x d 4 , such that 1. 2. 3. 4. pk ., for

Dt, x iŽ a. s 0 for a g A, Dt, x iŽ x i . s t, Dt, x iŽ x j . s 0 for j / i, Dt, x iŽ M Ž p1 , . . . , pk .. s Ý kjs1 M Ž p1 , . . . , pjy1 , Dt, x iŽ pj ., pjq1 , . . . , M g S and k s k Ž M ..

Observe that condition 4 in the definition generalizes the formula for the derivation of a product. We continue this section by showing that polarizations satisfy certain chain rules. The proofs of these chain rules will follow by induction on the elements in A x 1 , . . . , x d 4 . PROPOSITION 3.2. Let p g A x 4 and q g A y4 . We may consider the element evalŽ p; x ¤ q . as an element of A y4 . Moreo¨ er, the polarization of this element by Du, y is gi¨ en by Du , y eval Ž p; x ¤ q . s eval Ž Dt , x p; t ¤ Du , y Ž q . , x ¤ q . . Proof. The proof is by induction on p g A x 4 . It is easy to check that it holds for p s a g A and for p s x. Moreover, both sides are linear in p.

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APOLARITY AND GENERIC CANONICAL FORMS

What remains to show is the induction step. Namely, given that the result holds for p1 , . . . , pk g A x 4 , then it also holds for any p s M Ž p1 , . . . , pk ., where M is a multilinear form in S such that k Ž M . s k. We have that Du , y eval Ž M Ž p1 , . . . , pk . ; x ¤ q . s Du , y Ž M Ž eval Ž p1 ; x ¤ q . , . . . , eval Ž pk ; x ¤ q . . . k

s

Ý M Ž eval Ž p1 ; x ¤ q . , . . . , Du , y Ž eval Ž pi ; x ¤ q . . , is1

. . . ,eval Ž pk ; x ¤ q . . k

s

Ý M Ž eval Ž p1 ; x ¤ q . , . . . , eval Ž Dt , x pi ; t ¤ Du , y Ž q . , x ¤ q . , is1

. . . ,eval Ž pk ; x ¤ q . . k

s

Ý eval Ž M Ž p1 , . . . , Dt , x pi , . . . , pk . ; t ¤ Du , y Ž q . , x ¤ q . is1

s eval Ž Dt , x M Ž p1 , . . . , pk . ; t ¤ Du , y Ž q . , x ¤ q . s eval Ž Dt , x p; t ¤ Du , y Ž q . , x ¤ q . . Hence the induction is complete. We say that a function ␾ from the complex numbers ⺓ to the vector space A is differentiable if for all ␣ g ⺓ the limit lim

␾ Ž ␣ q h. y ␾ Ž ␣ . h

hª0

exists. When this limit exists, we denote it with Ž ⭸r⭸␣ . ␾ Ž ␣ . s ␾ ⬘Ž ␣ .. Let M be a multilinear form in S , with k s k Ž M .. Let ␾ i , for i s 1, . . . , k, be differentiable functions from ⺓ to A. We claim that M Ž ␾ 1 , . . . , ␾ k . is also a differentiable function from ⺓ to A. This is easy to check with the following computation. M Ž ␾ 1Ž ␣ q h . , . . . , ␾ k Ž ␣ q h . . y M Ž ␾ 1Ž ␣ . , . . . , ␾ k Ž ␣ . . h k

s

ÝM is1

ž

␾ 1 Ž ␣ q h . , . . . , ␾ iy1 Ž ␣ q h . ,

␾i Ž ␣ q h . y ␾i Ž ␣ . h

,

␾ iq1 Ž ␣ . , . . . , ␾ k Ž ␣ . .

/

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RICHARD EHRENBORG

Since M is continuous the above expression converges when h ª 0, and the limit is k

Ý M Ž ␾ 1Ž ␣ . , . . . , ␾iy1Ž ␣ . , ␾iX Ž ␣ . , ␾iq1Ž ␣ . , . . . , ␾ k Ž ␣ . . . is1

Observe that this expression is like the derivation of a product. We also have the following chain rule for polarizations. PROPOSITION 3.3. Let p g A x 1 , . . . , x d 4 and let ␾ i be a differentiable function from ⺓ to A for i s 1, . . . , d. Then

⭸ ⭸␣

eval Ž p; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . d

s

Ý eval Ž Dt , x

i

p; ␾ iX Ž ␣ . , ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . .

is1

Proof. Just as in the proof of Proposition 3.2, we proceed by induction on the elements p of A x 1 , . . . , x d 4 . It is easy to prove the basis for induction, that is, when p s a g A and p s x i . Notice that, as in the previous proposition, both sides are linear in p. It remains to show the induction step. Given that the result holds for p1 , . . . , pk g A x 4 , then it also holds for any p s M Ž p1 , . . . , pk ., where M is a multilinear form in S such that k Ž M . s k. Then

⭸ ⭸␣

eval Ž M Ž p1 , . . . , pk . ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . s

⭸ ⭸␣

M Ž eval Ž p1 ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . , . . . ,eval Ž pk ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . .

k

s

ÝM js1

ž

eval Ž p1 ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . ,

...,

⭸ ⭸␣

eval Ž pj ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . ,

. . . , eval Ž pk ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . .

/

APOLARITY AND GENERIC CANONICAL FORMS k

s

173

d

Ý Ý M ž eval Ž p1 ; ␾ 1Ž ␣ . , . . . , ␾d Ž ␣ . . ,

js1 is1

. . . , eval Ž Dt , x i pj ; ␾ iX Ž ␣ . , ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . , . . . , eval Ž pk ; ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . . d

s

k

Ý Ý eval Ž M Ž p1 , . . . , Dt , x

/

i

pj , . . . , pk . ; ␾ iX Ž ␣ . , ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ . .

i

pj , . . . , pk . ; ␾ iX Ž ␣ . , ␾ 1 Ž ␣ . , . . . , ␾ d Ž ␣ .

is1 js1 d

s

k

ž

Ý eval Ý M Ž p1 , . . . , Dt , x is1

js1

/

d

s

Ý eval Ž Dt , x M Ž p1 , . . . , pk . ; ␾iX Ž ␣ . , ␾ 1Ž ␣ . , . . . , ␾d Ž ␣ . . . i

is1

Here the induction is complete. When A is a commutative algebra, we may express the polarizations in terms of classical derivations. That A is a commutative algebra means that S consists of one bilinear form that is symmetric Žcommutative., fulfills the associative law, and has a unit element 1. For an algebra we denote this unique bilinear form by ⭈. DEFINITION 3.4. For a commutative algebra A we define the derivation Dx i : A x 1 , . . . , x d 4 ª A x 1 , . . . , x d 4 to be the linear map satisfying 1. 2. 3. 4.

Dx iŽ a. s 0 for a g A, Dx iŽ x i . s 1, Dx iŽ x j . s 0 for j / i, Dx iŽ p ⭈ q . s Dx iŽ p . ⭈ q q p ⭈ Dx iŽ q ..

PROPOSITION 3.5. Let A be a commutati¨ e algebra. Then for any element p g A x 1 , . . . , x d 4 and for a, a1 , . . . , a d g A we ha¨ e that eval Ž Dt , x i p; a, a1 , . . . , a d . s a ⭈ eval Ž Dx i p; a1 , . . . , a d . . We omit the proof, since it is a straightforward induction argument.

4. HOMOGENEOUS POLYNOMIALS AND APOLARITY DEFINITION 4.1. Let V, W1 , . . . , Wd be linear subspaces of A. We say that an element p in the set A x 1 , . . . , x d 4 is homogeneous with respect to the linear spaces V, W1 , . . . , Wd , if for all w 1 g W1 , . . . , wd g Wd we have evalŽ p; w 1 , . . . , wd . g V.

174

RICHARD EHRENBORG

PROPOSITION 4.2. Let V be a subspace of A of dimension n and let Wj be a subspace of A of dimension n j for j s 1, . . . , d. Let p g A x 1 , . . . , x d 4 be homogeneous with respect to the linear spaces V, W1 , . . . , Wd . Let u1 , . . . , u n be a basis of V and z j, 1 , . . . , z j, n j be a basis of Wj . Par s Ž j, i . : 1 F j F d, 1 F i F n j 4 . Thus for k g Par, z k is a base ¨ ector in Wj for some index j. Now there exist polynomials ␾ i g ⺓w ␣ k x k g Par in the ¨ ariables ␣ j, i such that n1

n

Ý ␾i Ž ␣ k . kgPar u i s eval is1

ž

p;

nd

Ý ␣ 1, i z1, i , . . . , Ý ␣ d , i z d , i is1

is1

/

.

Proof. The right-hand side consists of multilinear forms. Hence we may use the multilinear property and linearly expand the expression. In doing this we obtain a linear combination of elements of the form eval Ž p; z1 , m 1 , . . . , z d , m d . , where 1 F m j F n j . Moreover, the coefficients in this linear combination will be polynomials in the ␣ ’s. Since p is homogeneous, we know that each of the elements evalŽ p; z1, m 1, . . . , z d, m d . lies in V, and thus can be expanded in the basis u1 , . . . , u n . By composing these two expansions the result follows. PROPOSITION 4.3. Assume that the polynomial p g A x 1 , . . . , x d 4 is homogeneous with respect to the linear spaces V, W1 , . . . , Wd . Then the polynomial Dt, x iŽ p . is homogeneous with respect to the linear spaces V, Wi , W1 , . . . , Wd . Proof. Since p is homogeneous with respect to the linear spaces V, W1 , . . . , Wd , we know that the element evalŽ p; w 1 , . . . , wi q ␣ ⭈ w, . . . , wd ., lies in V, where ␣ g ⺓, w g Wi , and wj g Wj for j s 1, . . . , d. Take the partial derivative in the variable ␣ of this element. The derivative will also take values in V. By Proposition 3.3 the derivative is equal to

⭸ ⭸␣

eval Ž p; w 1 , . . . , wi q ␣ ⭈ w, . . . , wd . s eval Dt , x i p ;

ž

⭸ ⭸␣

Ž wi q ␣ ⭈ w . , w1 , . . . , wi q ␣ ⭈ w, . . . , wd

s eval Ž Dt , x i p; w, w 1 , . . . , wi q ␣ ⭈ w, . . . , wd . . Now by setting ␣ s 0 the conclusion follows. We now introduce the concept of apolarity in its general setting.

/

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APOLARITY AND GENERIC CANONICAL FORMS

DEFINITION 4.4. Let V and W be vector spaces and let f be a linear map from W to V. We say L g V * is apolar to f if for all w g W

² L f Ž w .: s 0. One important class of linear maps on which we consider the apolarity condition is given by the following. PROPOSITION 4.5. Let p g A x 1 , . . . , x d 4 and let a1 , . . . , a d g A. Then the following map from A to A is linear y ¬ eval Ž Dt , x i p; y, a1 , . . . , a d . . Proof. The proof is by induction on p g A x 1 , . . . , x d 4 . When p s a g A then Dt, x i p s 0, so the map is the zero map, which is linear. When p s x j then Dt, x i p s ␦ i, j t. So y ¬ ␦ i, j y, which is linear. Hence the two base cases are done. The linear combination of two linear maps is a linear map. The remaining case to consider is when p s M Ž p1 , . . . , pk ., where M g S and k s k Ž M .. Assume it holds for p1 , . . . , pk . Then we have that eval Ž Dt , x i M Ž p1 , . . . , pk . ; y, a1 , . . . , a d . k

s eval

ž

Ý M Ž p 1 , . . . , Dt , x

i

pj , . . . , pk . ; y, a1 , . . . , a d

js1

/

k

s

Ý M Ž eval Ž p1 ; a1 , . . . , a d . , . . . , eval Ž Dt , x

i

pj ; y, a1 , . . . , a d . ,

js1

. . . ,eval Ž pk ; a1 , . . . , a d . . . Each term in this expression is a linear map. Since the sum of linear maps is a linear map, the proof is done.

5. GENERIC CANONICAL FORMS DEFINITION 5.1. Let V be a finite dimensional linear space. We say that a generic element ¨ g V has a property P, if the set of all elements in V that has this property forms a dense set in V, where V has the Euclidean topology. THEOREM 5.2. Let V, W1 , . . . , Wd be finite dimensional linear subspaces of the algebra A. Let p in A x 1 , . . . , x d 4 be homogeneous with respect to the linear spaces V, W1 , . . . , Wd . A generic element ¨ g V can be written in the form ¨ s eval Ž p; w 1 , . . . , wd .

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RICHARD EHRENBORG

for some w 1 , . . . , wd if and only if there exist wX1 , . . . , wdX so that there is no nonzero dual element in V * which is apolar to the linear map y j ¬ eval Ž Dt , x j p; y j , wX1 , . . . , wdX . relati¨ e to Wj , for all 1 F j F d. Observe that Proposition 4.5 shows that the map y j ¬ evalŽ Dt, x j p; y j , wX1 , . . . , wdX . is linear, and that Proposition 4.3 guarantees that it maps the linear space Wj into V. The proof of the theorem requires the following two propositions. Proofs of these two propositions may be found in w3x. Let ⺓ Ž x 1 , . . . , x q . be the field of all algebraical functions in the variables x 1 , . . . , x q . P ROPOSITION 5.3. Let p 1Ž x 1 , . . . , x q ., . . . , p r Ž x 1 , . . . , x q . g ⺓ Ž x 1 , . . . , x q . , where r F q. Then the algebraic functions p1 , . . . , pr are algebraically independent if and only if the matrix

⭸ pi

ž / ⭸ xj

1FiFr , 1FjFq

has full rank. P ROPOSITION 5.4. Let p 1Ž x 1 , . . . , x q ., . . . , p r Ž x 1 , . . . , x q . ⺓ Ž x 1 , . . . , x q . , where r F q. Let P: ⺓ q ª ⺓ r be defined by

g

P Ž x 1 , . . . , x q . s Ž p1 Ž x 1 , . . . , x q . , . . . , pr Ž x 1 , . . . , x q . . . Then the algebraic functions p1 , . . . , pr are algebraically independent if and only if the range of the map P is dense in ⺓ r. Proof of Theorem 5.2. Let dimŽ V . s n and dimŽWj . s n j for j s 1, . . . , d. Let u1 , . . . , u n be a basis for V, and similarly let z j, 1 , . . . , z j, n j be a basis for Wj . Thus an element wj g Wj can be written in the form nj

wj s

Ý ␣ j, i z j, i , is1

where ␣ j, i g ⺓. We will call the coefficients ␣ j, i parameters. Set Par s Ž j, i . : 1 F j F d, 1 F i F n j 4 . Thus a parameter is of the form ␣ k , where k g Par. Observe that the number of parameters is
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APOLARITY AND GENERIC CANONICAL FORMS

parameters on the right hand side, we obtain the inequality n F
n

Ý ␾i Ž ␣ k . kgPar u i s eval is1

ž

nd

Ý ␣ 1, i z1, i , . . . , Ý ␣ d , i z d , i

p;

is1

is1

/

,

where ␾ i are polynomials for i s 1, . . . , n. Consider the map ⌽: ⺓ Par ª ⺓ n defined by ⌽ Ž Ž ␣ k . kgPar . s Ž ␾ i Ž ␣ k . kgPar . 1FiFn , where the coordinates of ⺓ Par are indexed by the set Par. The assumption is that the range of the map ⌽ is dense in ⺓ n. By Proposition 5.4 we infer that the n polynomials ␾ i are algebraically independent. Hence, by Proposition 5.3, the matrix

⭸␾ i

ž / ⭸␣ k

Ž 1. 1FiFn , kgP ar

has full rank, where rows are indexed by i and the columns by the set Par. Since the matrix Ž1. has full rank, we can choose values for the parameters such that the matrix Ž1. still has full rank. Denote these values we choose for the parameters by ␥ k for k g Par. Let wjX s

nj

Ý ␥ j, i z j, i . is1

Thus wjX g Wj . Moreover, we know that the matrix

ž

⭸␾ i ⭸␣ k

/

␣ m s ␥ m 1F iFn , kgPar

has full rank. Hence the columns of the matrix span the linear space ⺓ n. But ⺓ n is isomorphic to V. Via this isomorphism we get

⭸␾ i

ž / ⭸␣ k

n

¬ 1F iFn

Ý is1

⭸␾ i ⭸␣ k

ui s

⭸¨ ⭸␣ k

.

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RICHARD EHRENBORG

Thus, the elements

⭸¨ ⭸␣ k

␣ ms␥ m

span the linear space V. Hence there is no nonzero functional L g V * such that

⭸¨

¦

L

⭸␣ k

;

s0

␣ ms␥ m

for all k g Par. Each of the parameters will only occur in one of the vectors w 1 , . . . , wd . The parameter ␣ j, i occurs only in wj . In particular, we have

⭸ wj ⭸␣ j, i

s z j, i .

Hence by the chain rule, Proposition 3.3, we conclude that

⭸¨ ⭸␣ j, i

s

⭸ ⭸␣ j, i

eval Ž p; w 1 , . . . , wd .

s eval Dt , x j p;

ž

⭸ wj ⭸␣ j, i

, w 1 , . . . , wd

/

s eval Ž Dt , x j p; z j, i , w 1 , . . . , wd . . Observe that eval Ž Dt , x j p; z j, i , w 1 , . . . , wd .

␣ ms␥ m

s eval Ž Dt , x j p; z j, i , wX1 , . . . , wdX . .

Thus we can write our condition as follows: there is no nonzero functional L g V * such that

¦L

eval Ž Dt , x j p; z j, i , wX1 , . . . , wdX . s 0

;

for all 1 F j F d and 1 F i F n j . Observe that the above expression is linear in z j, i and recall that the elements z j, 1 , . . . , z j, n j form a basis for Wj . Hence the statement above is equivalent to the condition that there is no nonzero functional L g V * such that

¦L

eval Ž Dt , x j p; y j , wX1 , . . . , wdX . s 0

;

APOLARITY AND GENERIC CANONICAL FORMS

179

for all 1 F j F d and for all y j g Wj . Thus we have proven that there is no nonzero element in V * apolar to all the maps y j ¬ eval Ž Dt , x j p; y j , wX1 , . . . , wdX . relative to Wj for j s 1, . . . , s. This proves the necessary condition of the theorem. To prove the sufficient condition, we can trace the equivalences in the necessary part in opposite direction. We begin by assuming that there is no nonzero element in V * apolar to all the maps y j ¬ eval Ž Dt , x j p; y j , wX1 , . . . , wdX . relative to Wj for j s 1, . . . , d. This implies that the elements eval Ž Dt , x j p; z j, i , wX1 , . . . , wdX . , where 1FjFd and 1FiFn j , span the linear space V. Hence n1 q ⭈⭈⭈qn d elements span a linear space of dimension n. Thus n F n1 q ⭈⭈⭈ qn d . By the identities above, we can rewrite the above elements, and by using the isomorphism between V and ⺓ n we get that the matrix

ž

⭸␾ i ⭸␣ k

/

␣ m s ␥ m 1F iFn , kgPar

has rank d. Since n F n1 q ⭈⭈⭈ qn d the above matrix has full rank. Thus the matrix

⭸␾ i

ž / ⭸␣ k

, 1FiFn , kgPar

where we remove the values of the ␤ ’s, cannot have lower rank. But the rank cannot increase so the last matrix has full rank also. By Proposition 5.3 we know that the polynomials ␾ 1 , . . . , ␾ d are algebraically independent. Proposition 5.4 implies that the range of the map ⌽: ⺓ Par ª ⺓ d is dense and the theorem follows. When the S-algebra is a commutative and associative algebra with a unit element, then the linear maps in the statement of Theorem 5.2 reduce to y j ¬ y j ⭈ eval Ž Dx j p; wX1 , . . . , wdX . ,

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where Dx j is the derivation in the variable x j ; see Proposition 3.5. Moreover, if the algebra A is the algebra of polynomials with complex coefficients, and Wj is the linear space of homogeneous polynomials of degree d j , then Theorem 5.2 reduces to the result presented in w3x. 6. THE ESSENTIAL RANKS OF MULTI-DIMENSIONAL MATRICES Let W1 , . . . , Wd be vector spaces over the complex numbers such that dimŽWi . s n i . Consider the linear space V s W1 m W2 m ⭈⭈⭈ m Wd . Assume that z j, 1 , . . . , z j, n j is a basis for Wj . Then an element ¨ g V may be written as n1

¨s

n2

Ý Ý

i1s1 i 2s1

nd

⭈⭈⭈

Ý

i ds1

a i1 , i 2 , . . . , i d ⭈ z1, i1 m z 2, i 2 m ⭈⭈⭈ m z d , i d .

Thus the element ¨ corresponds to an n1 = n 2 = ⭈⭈⭈ = n d matrix Ž a i , i , . . . , i .1 F i F n . Hence V is isomorphic to the linear space of n1 = 1 2 d j j n 2 = ⭈⭈⭈ = n d matrices whose entries are complex numbers. We will now extend the idea of rank of a two dimensional matrix to the multi-dimensional case. DEFINITION 6.1. An element ¨ g W1 m ⭈⭈⭈ m Wd has rank k if k is the smallest integer m such that there exist w 1, 1 , . . . , w 1, m g W1 , . . . ,wd, 1 , . . . , wd, m g Wd , such that m

¨s

Ý w1, i m ⭈⭈⭈ m wd , i . is1

A question to study is: What is the maximal rank of the elements ¨ in W1 m ⭈⭈⭈ m Wd? Clearly an upper bound for the maximal rank is ŽmaxŽ n1 , . . . , n d ..y1 ⭈ Ł djs1 n j . For instance, when d s 2 it is well-known that this bound is reached, namely the maximal rank of W1 m W2 is minŽ n1 , n 2 .. Also, when d s 3 and n1 s n 2 s n 3 s 2 then the maximal rank of W1 m W2 m W3 is 3; see w5x. An example of an element that has rank 3 is z1, 2 m z 2, 1 m z 3, 1 q z1, 1 m z 2, 2 m z 3, 1 q z1, 1 m z 2, 1 m z 3, 2 . Instead of considering the maximal rank, we will continue with a related question. DEFINITION 6.2. The linear space W1 m ⭈⭈⭈ m Wd has essential rank k if k is the smallest integer m such that the set of elements ¨ in W1 m ⭈⭈⭈ m Wd of rank at most m forms a dense set in the Euclidean topology.

APOLARITY AND GENERIC CANONICAL FORMS

181

Observe that the maximal rank of W1 m ⭈⭈⭈ m Wd might be different from the essential rank of W1 m ⭈⭈⭈ m Wd . For instance, when n1 s n 2 s n 3 s 2, the essential rank of W1 m W2 m W3 is 2, which is different from the maximal rank 3. For more on this last example, see w5x. To find upper bounds for the essential rank of the linear space W1 m ⭈⭈⭈ m Wd we will consider the combinatorial problem of rook coverings. Let A1 , . . . , A d be finite sets and let A1 = ⭈⭈⭈ = A d be the Cartesian product of these sets. DEFINITION 6.3. A rook covering of A1 = ⭈⭈⭈ = A d is a subset R of A1 = ⭈⭈⭈ = A d such that for all Ž a1 , . . . , a d . in A1 = ⭈⭈⭈ = A d there exist Ž r 1 , . . . , r d . g R such that Ž a1 , . . . , a d . and Ž r 1 , . . . , r d . differ in at most one place. That is,

 i : ai / ri 4 F 1. We call a rook covering R perfect if for all Ž a1 , . . . , a d . g A1 = ⭈⭈⭈ = A d there exists a unique Ž r 1 , . . . , r d . g R with the above conditions. Let n1 , . . . , n d be positive integers. We will say that there is a rook covering of n1 = ⭈⭈⭈ = n d of cardinality N, if there is a rook covering of A1 = ⭈⭈⭈ = A d of cardinality N, where A i has size n i for i s 1, . . . , d. The problem of finding small rook coverings is a well studied problem, see w8, 13, 18x. PROPOSITION 6.4. Let Wj be a ¨ ector space o¨ er the complex numbers of dimension n j for j s 1, . . . , d. If there is a rook co¨ ering of n1 = ⭈⭈⭈ = n d of cardinality N, then the essential rank of V s W1 m ⭈⭈⭈ m Wd is less than or equal to N. Proof. Let A be the vector space defined as A s W1 [ ⭈⭈⭈ [ Wd [ V . Since A is finite dimensional, let the topology of A be the Euclidean. Define M :^A =` ⭈⭈⭈ = A _ª A d

to be a multilinear form, such that M Ž a1 , . . . , a d . s

½

a1 m ⭈⭈⭈ m a d

if a j g Wj ,

0

otherwise,

and extend M by linearity. It is easy to see that M is continuous.

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Clearly the polynomial p g A x 1, 1 , . . . , x 1, N , . . . , x d, 1 , . . . , x d, N 4 defined by N

ps

Ý M Ž x 1, i , . . . , x d , i . is1

is homogeneous with respect to V , W1 , . . . , W1 , . . . , Wd , . . . , Wd .

^ ` _

^ ` _

N

N

We would like to show that a generic element of V can be written in the form eval Ž p; w 1, 1 , . . . , w 1, N , . . . , wd , 1 , . . . , wd , N . ,

Ž 2.

where wj, i g Wj . Observe that D t, x j, iŽ p . s M Ž x 1, i , . . . , x jy 1, i , t, x jq1, i , . . . , x d, i .. By Theorem 5.2 it is enough to prove that there exist wj,X i g Wj for j s 1, . . . , d and i s 1, . . . , N such that there is no nonzero dual element L g V * apolar to all the linear maps X X y j, i ¬ M Ž wX1, i , . . . , wXjy1, i , y j, i , wjq1, i , . . . , wd , i . ,

Ž 3.

where y j, i g Wj . In order to do this, let z j, 1 , . . . , z j, n j be a basis for Wj . Similarly let zUj, 1 , . . . , zUj, n j be the dual basis for WjU . That is, ² zUj, i < z j, k : s ␦ i, k . Let A j be the set  1, . . . , n j 4 . Let R be a rook covering of A1 = ⭈⭈⭈ = A d of cardinality N. Thus let R s  Ž r 1, i , . . . , r d , i . : 1 F i F N 4 . Choose wXj, i s z j, r j, i . Assume that L in V * is apolar to all the linear maps in Ž3.. We can write the dual element L in terms of the dual basis. Ls

Ý

Ž i 1 , . . . , i d .gA 1 = ⭈⭈⭈ =A d

␤i1 , . . . , i d ⭈ zU1, i1 m ⭈⭈⭈ m zUd , i d .

Consider an element Ž i1 , . . . , i d . g A1 = ⭈⭈⭈ = A d . Since R is a rook covering, there is an element Ž r 1, k , . . . , r d, k . g R that differs in at most one coordinate from Ž i1 , . . . , i d .. Let the coordinate where Ž i1 , . . . , i d . and Ž r 1, k , . . . , r d, k . differ be j. ŽIf Ž i1 , . . . , i d . s Ž r 1, k , . . . , r d, k . then choose j arbitrarily. . Thus we know that L is apolar to the linear map y j, k ¬ M Ž z1, r 1 , k , . . . , z jy1, r jy 1 , k , y j, k , z jq1, r jq 1 , k , . . . , z d , r d , k . .

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Let y j, k take the value of z j, i j . Hence 0 s L M Ž z1, r 1 , k , . . . , z jy1, r jy 1 , k , z j, i j , z jq1, r jq 1 , k , . . . , z d , r d , k .

¦ s¦L z

1, i 1

; ;

m ⭈⭈⭈ m z jy1, i jy 1 m z j, i j m z jq1, i jq 1 m ⭈⭈⭈ m z d , i d

s ␤i1 , . . . , i d . We conclude that all coefficients of L vanish. Thus we know that L s 0. Theorem 5.2 implies that a generic element of V can be written in the form Ž2.. Thus the linear space V s W1 m ⭈⭈⭈ m Wd has essential rank less than or equal to N. The smallest number of rooks covering the set n = n = n is n2r2 . This was stated as a problem in the Soviet Olympiad 1971. See also w2, Problem 39x. Thus we obtain: COROLLARY 6.5. Let W be a ¨ ector space o¨ er the complex numbers of dimension n. Then W m3 has essential rank less than or equal to n2r2 . An error correcting code that corrects one error may be viewed as a perfect rook covering. Such a code is the Hamming codes w11x. They have the following parameters. Let q be a prime power, and let d s Ž q k y 1.rŽ q y 1., where k is a positive integer. Then there is a perfect rook covering of q d s q = ⭈⭈⭈ = q

^ ` _ d

of size q dy k . We then obtain: COROLLARY 6.6. Let W be a ¨ ector space o¨ er the complex numbers of dimension q, where q is a prime power. Let d s Ž q k y 1.rŽ q y 1., where k be a positi¨ e integer. Then W md has essential rank less than or equal to q dy k . PROPOSITION 6.7. If there is a rook co¨ ering of n1 = ⭈⭈⭈ = n d using N rooks, then there is a rook co¨ ering of Ž m ⭈ n1 . = ⭈⭈⭈ = Ž m ⭈ n d . using m dy 1 ⭈ N rooks. Proof. Let A i be a set of size n i and let Bi s A i =  0, 1, . . . , m y 14 . Let R be a rook covering of A1 = ⭈⭈⭈ = A d of cardinality N. We would like to find a rook covering of the set B1 = ⭈⭈⭈ = Bd . Consider the set R⬘ defined as

 Ž Ž a1 , p1 . , . . . , Ž a d , pd . . g B1 = ⭈⭈⭈ = Bd : Ž a1 , . . . , a d . g R, p1 q ⭈⭈⭈ qpd ' 0 Ž mod m . 4 .

184

RICHARD EHRENBORG

It is a direct verification that R⬘ is a rook covering of B1 = ⭈⭈⭈ = Bd with cardinality m dy 1 ⭈ N. The linear analogue to Proposition 6.7 is: PROPOSITION 6.8. Let U, W1 , . . . , Wd be ¨ ector spaces o¨ er the complex numbers of dimensions m, n1 , . . . , n d , respecti¨ ely. If the linear space W1 m ⭈⭈⭈ m Wd has essential rank N, then the linear space ŽW1 m U . m ⭈⭈⭈ m ŽWd m U . has essential rank at most m dy 1 ⭈ N. Proof. Let z j, 1 , . . . , z j, n j be a basis of Wj , and let zUj, 1 , . . . , zUj, n j be the dual basis of WjU . Let I denote the set  1, . . . , n14 = ⭈⭈⭈ =  1, . . . , n d 4 . We know that a generic element of W1 m ⭈⭈⭈ m Wd can be written in the form N

Ý w1, i m ⭈⭈⭈ m wd , i , is1

where wj, i g Wj . By Theorem 5.2 this canonical form implies that there exist wj,X i g Wj , 1 F j F d and 1 F i F N, such that there is no nonzero element in ŽW1 m ⭈⭈⭈ m Wd .* apolar to all the maps t j, i ¬ wX1, i m ⭈⭈⭈ m t j, i m ⭈⭈⭈ m wdX , i , where t j, i g Wj . Hence the images of these linear maps span the space W1 m⭈⭈⭈m Wd . This is equivalent to the statement that for all Ž i1 , . . . , i d . g I there exist y j, i g Wj for 1 F j F d and 1 F i F N, such that d

N

Ý Ý wX1, i m ⭈⭈⭈ m yj, i m ⭈⭈⭈ m wdX , i s z i

1

m ⭈⭈⭈ m z i d .

Ž 4.

js1 is1

Let u1 , . . . , u m be a basis of U. Consider the following subset of  0, 1, . . . ,m y 14 d. d

P s  Ž p1 , . . . , pd . g  0, 1, . . . , m y 1 4 : p1 q ⭈⭈⭈ qpd ' 0 Ž mod m . 4 . The cardinality of P is m dy 1. Define the elements ¨ Xj, i, p g Wj m U, where j s 1, . . . , d, i s 1, . . . , N, and p g P, by X

X

¨ j, i , p s wj, i m u p j .

Thus there are d ⭈ N ⭈ m dy 1 such elements. Consider now the linear maps t j, i , p ¬ ¨ X1 , i , p m ⭈⭈⭈ m t j, i , p m ⭈⭈⭈ m ¨ Xd , i , p ,

Ž 5.

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APOLARITY AND GENERIC CANONICAL FORMS

where t j, i, p g Wj m U. We would like to show that the images of these linear maps span the linear space ŽW1 m U . m ⭈⭈⭈ m ŽWd m U .. Choose Ž i1 , . . . , i d . g I and Ž q1 , . . . , q d . g  0, 1, . . . , m y 14 d. This choice corresponds to the basis element

Ž zi

1

m u q1 . m ⭈⭈⭈ m Ž z i d m u q d .

of ŽW1 m U . m ⭈⭈⭈ m ŽWd m U .. As observed before, we can find y j, i g Wj for 1 F j F d and 1 F i F N, such that Eq. Ž4. is satisfied. Let p j s Ž p1j , . . . , pdj . be the element of P such that

pkj s

½

qk

if k / j,

qk y Ž q1 q ⭈⭈⭈ qqd . Ž mod m .

if k s j.

Thus q and p j only differ in the jth coordinate. Let $

y j, i s y j, i m u q j .

$

The element ¨ X1, i, p j m ⭈⭈⭈ m y j, i m ⭈⭈⭈ m ¨ Xd, i, p j lies in the image of one of the maps in Ž5.. Consider now the sum of these elements. N

d

Ý Ý ¨ X1, i , p

$

j

m ⭈⭈⭈ m y j, i m ⭈⭈⭈ m ¨ Xd , i , p j

is1 js1 N

s

d

Ý Ý Ž wX1, i m u p . m ⭈⭈⭈ m Ž yj, i m u q . m ⭈⭈⭈ m Ž wdX , i m u p . j 1

j

j d

is1 js1 N

s

d

Ý Ý Ž wX1, i m u q . m ⭈⭈⭈ m Ž yj, i m u q . m ⭈⭈⭈ m Ž wdX , i m u q . . 1

j

d

is1 js1

Observe that the two linear spaces ŽW1 m U . m ⭈⭈⭈ m ŽWd m U . and W1 m ⭈⭈⭈ m Wd m U md are naturally isomorphic. Let ⌽ be this isomorphism, that is, ⌽ : W1 m ⭈⭈⭈ m Wd m U md ª Ž W1 m U . m ⭈⭈⭈ m Ž Wd m U . .

186

RICHARD EHRENBORG

View the isomorphism ⌽ as a reordering of the terms. Now, the above element in ŽW1 m U . m ⭈⭈⭈ m ŽWd m U . can be written as N

⌽

ž

d

Ý Ý wX1, i m ⭈⭈⭈ m yj, i m ⭈⭈⭈ m wdX , i m u q

1

m ⭈⭈⭈ m u q j m ⭈⭈⭈ m u q d

is1 js1 N

s⌽

žž

/

d

Ý Ý wX1, i m ⭈⭈⭈ m yj, i m ⭈⭈⭈ m wdX , i is1 js1

/

m u q1 m ⭈⭈⭈ m u q d

/

s ⌽ Ž z i1 m ⭈⭈⭈ m z i d m u q1 m ⭈⭈⭈ m u q d . s Ž z i1 m u q1 . m ⭈⭈⭈ m Ž z i d m u q d . . But this is the particular basis element we chose in ŽW1 m U . m ⭈⭈⭈ m ŽWd m U .. Hence we conclude that the images of the linear maps Ž5. span the space ŽW1 m U . m ⭈⭈⭈ m ŽWd m U .. By Theorem 5.2 it follows that the linear space ŽW1 m U . m ⭈⭈⭈ m ŽWd m U . has essential rank at most m dy 1 ⭈ N.

7. ESSENTIAL RANK OF SYMMETRIC AND SKEW-SYMMETRIC TENSORS Let W be a vector space over the complex numbers of dimension n. Recall that SymŽW . is the algebra of symmetric tensors over W. That is, SymŽW . is isomorphic to the algebra of polynomials in n variables with complex coefficients. Similarly, ExtŽW . is the exterior algebra on W. The product in SymŽW . is denoted by ⭈ and the product in ExtŽW . is denoted by n. Both these algebras are graded, and we may write Ext Ž W . s

Ext [ dG0

d

Ž W . and SymŽ W . s [ Sym d Ž W . . dG0

Observe that dim Ž Ext d Ž W . . s

n d

ž /

and

dim Ž Sym d Ž W . . s

n , d

¦;

where Ž nd . is the number of ways to select a subset of size d from an n-element set and where ² nd : is the number of ways to select a multisubset of cardinality d from an n-element set. We have that ² nd : s Ž nqddy1 .. Similar to Definition 6.1 we may define the rank of a symmetric respectively skew-symmetric tensor.

APOLARITY AND GENERIC CANONICAL FORMS

187

DEFINITION 7.1. An element ¨ in the linear space Ext d ŽW . has rank k if k is the smallest integer m such that there exist y j, i g W for 1 F j F d and 1 F i F m such that m

¨s

Ý y1, i n ⭈⭈⭈ n yd , i . is1

Similarly, an element ¨ in the linear space Sym d ŽW . has rank k if k is the smallest integer m such that there exist y j, i g W for 1 F j F d and 1 F i F m such that m

¨s

Ý y1, i

⭈⭈⭈ yd , i .

is1

When d s 2 the maximal ranks that occur in Ext 2 ŽW . and Sym 2 ŽW . are well-known. PROPOSITION 7.2. Let W be a ¨ ector space o¨ er the complex numbers of dimension n. Then the maximal rank of Ext 2 ŽW . is ? nr2@ and the maximal rank of Sym 2 ŽW . is u nr2v . Proof. The first result is a well-known result in exterior algebra; see, for example, w4x. An element in Sym 2 ŽW . corresponds to a polynomial homogeneous of degree 2 and in n variables. It is well-known from linear algebra that such a polynomial may be written as a sum of n squares Žor less. of linear forms. By the identity a2 q b 2 s Ž a q i ⭈ b . ⭈ Ž a y i ⭈ b ., we may combine 2 ⭈ ? nr2@ squares into ? nr2@ products. Similar to Definition 6.2 we have: DEFINITION 7.3. The linear space Ext d ŽW . ŽSym d ŽW .. has essential rank k if k is the smallest integer m such that the set of elements ¨ in Ext d ŽW . ŽSym d ŽW .. that has rank less than or equal to m form a dense set in the Euclidean topology. When dimŽW . s 6 it is known that Ext 3 ŽW . has essential rank 2; see w7x. Also when dimŽW . s 4 the essential rank of Sym 3 ŽW . is equal to 2; see, for instance, w3, Corollary 4.10x. For a set A let the set ŽdA . be the set of all subsets of cardinality d of A. Similarly, let ²dA : be the set of all multisubsets of cardinality d of A. Observe that we have A d

s

< A< d

ž / ž /

and

A d

< A< . d

¦; ¦ ; s

DEFINITION 7.4. A covering C of ŽdA . is a subset C of ŽdA . such that for all I g ŽdA . there exist J g C such that < I l J < G d y 1. Similarly, a

188

RICHARD EHRENBORG

co¨ ering C of ²dA : is a subset C of ²dA : such that for all I g ²dA : there exist J g C such that < I l J < G d y 1. We will Ž²dA :. say that there is a covering of Ždn . Ž²dn :. if there exists a covering of ŽdA . Ž²dA :., where cardinality of A is n. The problem of finding coverings of Žnd . as small as possible is called the lotto problem w14x. Similarly, the problem of finding small coverings of ²dn : is called the lotto problem with replacements. Similar to Proposition 6.4 we have: PROPOSITION 7.5. Let W be an n-dimensional ¨ ector space o¨ er the complex numbers ⺓. If there exists a co¨ ering of Ždn . of size N then the essential rank of Ext d ŽW . is less than or equal to N. If there exists a co¨ ering of ²dn : of size N then the essential rank of Sym d ŽW . is less than or equal to N. The proof is similar to the proof of Proposition 6.4 and thus is omitted. For the remaining part of this section we will consider the case when d s 3. DEFINITION 7.6. A Steiner triple system on a non-empty set A is a subset S of Ž3A . such that for all pairs P g Ž2A . there is a unique triple Q g S such that P : Q. A necessary and sufficient condition for a Steiner triple system to exist on a set A of cardinality n is that n ' 1, 3 mod 6. Observe also that the size of a Steiner triple system is 13 ⭈ Žn2 . s Ž n ⭈ Ž n y 1..r6. PROPOSITION 7.7. Assume that there exists a Steiner triple system on an n-set and on an m-set. Then there exists a co¨ ering of Žnq3 m . of size Ž n ⭈ Ž n y 1. q m ⭈ Ž m y 1..r6. Similarly, there exists a co¨ ering of ²nq3 m : of size Ž n ⭈ Ž n q 5. q m ⭈ Ž m q 5..r6. Proof. Assume that A1 and A 2 are two disjoint sets such that < A1 < s n and < A 2 < s m. Let S i be a Steiner triple system on A i for i s 1, 2. Then we claim that S1 j S2 is a covering of the set ŽA 1 j3 A 2 .. Assume that I g ŽA 1 j3 A 2 .. Then there is an index i such that < A i l I < G 2. Hence A i l I contains a pair P. Then we may find x g A i such that P j  x 4 g Si . For the multiset case, consider the set of multisets

  x, x, x 4 : x g A1 j A 2 4 j S1 j S2 , which has cardinality n q m q Ž n ⭈ Ž n y 1. q m ⭈ Ž m y 1..r6. It is easy to see that this is a covering of ²A 1 j3 A 2 :. COROLLARY 7.8.

Let n be an e¨ en positi¨ e integer.

If n ' 0, 8 mod 12 then there exists a co¨ ering of Žn3 . of size Ž n ⭈ Ž n y 2..r12 q 3. 䢇

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APOLARITY AND GENERIC CANONICAL FORMS

If n ' 2, 6 mod 12 then there exists a co¨ ering of Žn3 . of size Ž n ⭈ Ž n y 2..r12. 䢇

If n ' 4 mod 12 then there exists a co¨ ering of Žn3 . of size Ž n ⭈ Ž n y 2. q 4.r12. 䢇

If n ' 10 mod 12 then there exists a co¨ ering of Žn3 . of size Ž n ⭈ Ž n y 2. q 4.r12 q 1. 䢇

Proof. We will only prove the first statement. Since nr2 ' 0, 4 mod 6 we have that nr2 y 3 ' 1, 3 mod 6, and nr2 q 3 ' 1, 3 mod 6. Thus by Proposition 7.7, there exists a covering of size

Ž nr2y3 . ⭈ Ž nr2y4 . q Ž nr2q3 . ⭈ Ž nr2q2 . 6

s

n ⭈ Ž ny2 . 12

q3.

COROLLARY 7.9. Let W be an n-dimensional ¨ ector space o¨ er the complex numbers ⺓, where n is an e¨ en positi¨ e integer. If n ' 0, 8 mod 12 then the essential rank of Ext 3 ŽW . is less than or equal to Ž n ⭈ Ž n y 2..r12 q 3. 䢇

If n ' 2, 6 mod 12 then the essential rank of Ext 3 ŽW . is less than or equal to Ž n ⭈ Ž n y 2..r12. 䢇

If n ' 4 mod 12 then the essential rank of Ext 3 ŽW . is less than or equal to Ž n ⭈ Ž n y 2. q 4.r12. 䢇

If n ' 10 mod 12 then the essential rank of Ext 3 ŽW . is less than or equal to Ž n ⭈ Ž n y 2. q 4.r12 q 1. 䢇

In the case when the dimension of the linear space W is 8, Corollary 7.9 says that the essential rank of Ext 3 ŽW . is less than or equal to 7. But in fact, we do even better, as we will see in the next lemma. LEMMA 7.10. Let W be a linear space of dimension 8 o¨ er the complex number ⺓. Then the linear space V s Ext 3 ŽW . has essential rank less than or equal to 4. Proof. Let z1 , . . . , z 8 be a basis for W. Let zU1 , . . . , zU8 be the dual basis for W *. Then zUi n zUj n zUk , where 1 F i - j - k F 8, form a basis for V *.

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Choose wX1, i s z 2 iy1 , wX2, i s z 2 i , and wX3, i s z 2 iy2 y z 2 iq1 , for i s 1, 2, 3, 4 and where indices are counted modulo 8. Assume that L g V * is apolar to the 12 linear maps y 1 , i ¬ y 1, i n wX2, i n wX3, i s y 1, i n z 2 i n Ž z 2 iy2 y z 2 iq1 . , y 2 , i ¬ wX1, i n y 2, i n wX3, i s z 2 iy1 n y 2, i n Ž z 2 iy2 y z 2 iq1 . , y 3, i ¬ wX1, i n wX2, i n y 3, i s z 2 iy1 n z 2 i n y 3, i . We can write Ls

Ý

␣i , j, k4 ⭈ zUi n zUj n zUk .

1Fi-j-kF8

By using the fact that L is apolar to the third map with y 3, i s z k , we get that ␣2 iy1, 2 i, k4 s 0. Since L is apolar to the first map with y 1, i s z 2 iy2 , we have ␣2 iy2, 2 i, 2 iq14 s 0. Similarly, since L is apolar to the second map with y 2, i s z 2 iy2 , we have ␣2 iy2, 2 iy1, 2 iq14 s 0. Since L is apolar to the first map with y 1, i s z 2 iq2 , we get 0 s² L z 2 iq2 n z 2 i n Ž z 2 iy2 y z 2 iq1 .: s sign Ž 2 i q 2, 2 i , 2 i y 2 . ⭈ ␣2 iq2, 2 i , 2 iy24 y sign Ž 2 i q 2, 2 i , 2 i q 1 . ⭈ ␣2 iq2, 2 i , 2 iq14 s sign Ž 2 i q 2, 2 i , 2 i y 2 . ⭈ ␣2 iq2, 2 i , 2 iy24 . Again, use the first map with y 1, i s z 2 iq4 . 0 s² L z 2 iq4 n z 2 i n Ž z 2 iy2 y z 2 iq1 .: s sign Ž 2 i q 4, 2 i , 2 i y 2 . ⭈ ␣2 iq4, 2 i , 2 iy24 y sign Ž 2 i q 4, 2 i , 2 i q 1 . ⭈ ␣2 iq4, 2 i , 2 iq14 s ysign Ž 2 i q 4, 2 i , 2 i q 1 . ⭈ ␣2 iq4, 2 i , 2 iq14 . Use the first linear map with y 1, i s z 2 iq3 . 0 s² L z 2 iq3 n z 2 i n Ž z 2 iy2 y z 2 iq1 .: s sign Ž 2 i q 3, 2 i , 2 i y 2 . ⭈ ␣2 iq3, 2 i , 2 iy24 y sign Ž 2 i q 3, 2 i , 2 i q 1 . ⭈ ␣2 iq3, 2 i , 2 iq14 s sign Ž 2 i q 3, 2 i , 2 i y 2 . ⭈ ␣2 iq3, 2 i , 2 iy24 .

APOLARITY AND GENERIC CANONICAL FORMS

191

By symmetry, use the second map, with y 2, i s z 2 iy3 , y 2, i s z 2 iq3 , and y 2, i s z 2 iq4 , to conclude that ␣ 2 iy1, 2 iy3, 2 iq1 s 0, ␣ 2 iy1, 2 iq3, 2 iy2 s 0, and ␣ 2 iy1, 2 iq4, 2 iq1 s 0. It is easy to see that we have shown that all coefficients of L vanish, and thus L is equal to 0. By Theorem 5.2, Ý4is1wi, 1 n wi, 2 n wi, 3 is a generic canonical form for the linear space Ext 3 ŽW ., and thus the space has essential rank 4. The following corollary is straightforward to obtain. COROLLARY 7.11.

Let n be an e¨ en positi¨ e integer.

If n ' 0, 8 mod 12 then there exists a co¨ ering of ²n3 : of size Ž n ⭈ Ž n q 10..r12 q 3. 䢇

If n ' 2, 6 mod 12 then there exists a co¨ ering of ²n3 : of size Ž n ⭈ Ž n q 10..r12. 䢇

If n ' 4 mod 12 then there exists a co¨ ering of ²n3 : of size Ž n ⭈ Ž n q 10. q 4.r12. 䢇

If n ' 10 mod 12 then there exists a co¨ ering of ²n3 : of size Ž n ⭈ Ž n q 10. q 4.r12 q 1. 䢇

By Proposition 7.5 these upper bounds on covering ²3n : give us upper bounds on the essential rank of Sym 3 ŽW .. But we are able to press these bounds down a little by the following proposition. PROPOSITION 7.12 ŽR. E. Losonczy and J. Losonczy.. Let n and m be two positi¨ e integers such that there exists a Steiner triple system on sets of cardinalities n and m. Let W be a ¨ ector space o¨ er the complex number ⺓ of dimension n q m. Then the essential rank of Sym 3 ŽW . is less than or equal to n ⭈ Ž n y 1. q m ⭈ Ž m y 1. 6

q

n 3

q

m 3

.

Proof. Let A1 and A 2 be two disjoint sets of cardinalities n and m. Let A be the union of these two sets, that is, A s A1 j A 2 . Let Si be a Steiner triple system on A i . Hence < S1 < s Ž n ⭈ Ž n y 1..r6 and < S2 < s Ž m ⭈ Ž m y 1..r6. Let Ti be a set of three element subsets of A i such that A i s

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RICHARD EHRENBORG

D I g T i I. We would like the size of Ti to be as small as possible. Hence we may assume that < Ti < s < A i
Ý IgS 1

Ý

b1, I n b 2, I n b 3, I

q

igS 2

c1, I n c 2, I n c 3, I q

Ý

Ý

IgT 1

d1, I n d 2, I n d 3, I

IgT 2

is a generic canonical form for Sym 3 ŽW ., where a j, I , . . . , d j, I g W. Let  z i 4i g A j A be a basis for W and let  zUi 4i g A j A be the dual basis for W *. 1 2 1 2 Let aXj, I s z x , where x is the jth element in the triple I g S1. Similarly, X let bj, I s z x , where x is the jth element in the triple I g S2 . Now we choose cXj, I and dXj, I more carefully. For  x 1 , x 2 , x 34 s I g T1 let cX1, I s Ž z x q z x ., cX2, I s Ž z x q z x ., and cX3, I s Ž z x q z x .. The same pattern for 1 2 1 3 2 3 the dXj, I : for  x 1 , x 2 , x 34 s I g T2 let dX1, I s Ž z x 1 q z x 2 ., dX2, I s Ž z x 1 q z x 3 ., X and d 3, I s Ž z x 2 q z x 3 .. Let L g Sym 3 ŽW *. and assume that L is apolar to all the linear maps described by Theorem 5.2. Write L in terms of the dual basis Ls

Ý

␤i , j, k4 ⭈ zUi ⭈ zUj ⭈ zUk ,

 i , j, k 4

where the sum ranges over all three element multisubsets of A. We claim that for i, j g A1 and k g A we have that ␤i, j, k4 s 0. This is true since we may find a linear map y ¬ z i ⭈ z j ⭈ y, such that L is apolar to this linear map. By symmetry, for i, j g A 2 and k g A, we have ␤i, j, k4 s 0. We conclude that ␤I s 0 when I is a set, that is, when there are no repetitions in I. Hence any square free monomial in L has a vanishing coefficient. Let  i, j, k 4 gT1. Hence L is apolar to the map y¬ Ž z i qz j . ⭈ Ž z i qz k . ⭈y. For h g A we have that 0 s L Ž zi q z j . ⭈ Ž zi q zk . ⭈ zh

¦ s¦L

;

z i2 ⭈ z h q z i ⭈ z j ⭈ z h q z i ⭈ z k ⭈ z h q z j ⭈ z k ⭈ z h

;

s ␤i , i , h4 q ␤i , j, h4 q ␤i , k , h4 q ␤ j, k , h4 s ␤i , i , h4 . Since T1 contains triplets, which contain all the elements of A1 , we have that for all i g A1 and h g A the coefficient ␤i, i, h4 vanishes. By symmetry, and by working with T2 we obtain for i, h g A that ␤i, i, h4 s 0. Hence all the monomials in L vanish and we conclude that L is equal to zero. By Theorem 5.2 the conclusion follows.

APOLARITY AND GENERIC CANONICAL FORMS

193

COROLLARY 7.13. Let W be an n-dimensional ¨ ector space o¨ er the complex numbers ⺓, where n is an e¨ en positi¨ e integer. If n ' 0 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2..r12 q 3. If n ' 2 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2. q 4.r12 q 1. If n ' 4 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2..r12 q 1. If n ' 6 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2..r12. If n ' 8 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2. q 4.r12 q 4. If n ' 10 mod 12 then the essential rank of equal to Ž n ⭈ Ž n q 2..r12 q 2. 䢇

Sym 3 ŽW . is less than or

䢇

Sym 3 ŽW . is less than or

䢇

Sym 3 ŽW . is less than or

䢇

Sym 3 ŽW . is less than or

䢇

Sym 3 ŽW . is less than or

䢇

Sym 3 ŽW . is less than or

8. CONCLUDING REMARKS It would be interesting to know if Theorem 5.2 has more applications than the one presented in this article and in w3x. So far, all the examples seen have been over algebras such as the tensor algebra, the algebra of polynomials, and the exterior algebra. Thus the general notion of S-algebra has not been used yet. The problem of finding the essential rank of Ext d ŽW . has been considered recently by J. Losonczy w14x. He has found a geometrical interpretation of this essential rank. Moreover, he has worked out more upper bounds for the essential rank of Ext d ŽW .. In Section 7 we considered symmetric and skew-symmetric tensors. They correspond to sets and multisets, which are dual to each other. There may be a deeper relationship between these tensors, as the following conjecture suggests. CONJECTURE 8.1. Let n and d be non-negati¨ e integers and let m s n q d y 1. Let V and W be ¨ ector spaces o¨ er the complex numbers ⺓ of dimensions n and m. Is the essential rank of Sym d Ž V . equal to the essential rank of Ext d ŽW .? This hypothesis holds for small cases, for instance, when d F 2, and when n F 2. When n s 4 and d s 3 the results that Sym 3 Ž V . and Ext 3 ŽW . have essential rank 2 are known; see the comment after Definition 7.3. When n s 6 and d s 3 Corollary 7.13 implies that Sym 3 Ž V . has essential rank at most 4. By counting parameters, it is easy to conclude that the

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RICHARD EHRENBORG

essential rank of Sym 3 Ž V . is equal to 4. Lemma 7.10 says that the essential rank of Ext 3 ŽW . is at most 4. If this conjecture is true, it suggests a duality between the linear spaces Sym d Ž V . and Ext d ŽW .. A very vague hint for this duality is that dimŽSym d Ž V .. s dimŽExt d ŽW ... ACKNOWLEDGMENTS I thank Gian-Carlo Rota for introducing me to the area of apolarity and canonical forms. I also thank Gabor Hetyei, Jozsef Losonczy, and Margaret Readdy for many helpful ´ discussions and comments.

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