On the Schmidt and analytic ranks for trilinear forms
aa r X i v : . [ m a t h . AG ] F e b ON THE SCHMIDT AND ANALYTIC RANKS FORTRILINEAR FORMS
KARIM ADIPRASITO, DAVID KAZHDAN AND TAMAR ZIEGLER
Abstract.
We discuss relations between different notions of ranks for multi-linear forms. In particular we show that the Schmidt and the analytic ranksfor trilinear forms are essentially proportional. Introduction
In recent years there is growing interest in properties of polynomials which areindependent of the number of variables. To study such properties various notionswere introduced for measuring the complexity of polynomials. In this paper wecompare three notion of rank of polynomials P over a field k - the Schmidt rank r k ( P ), the slice rank s k ( P ) both defined for arbitrary fields k , and the analyticrank a k ( P ) defined for finite fields k = F q . Definition 1.1.
Let k be a field, V be a finite-dimensional k-vector space, V = V ( k ), let S k = L d ≥ S d k be the graded algebra of polynomial functions on V andlet P ∈ S d k .(1) The Schmidt rank r k ( P ) is the minimal number r such that P can bewritten in the form P = P ri =1 Q i R i , where Q i , R i ∈ S are polynomials on V of degrees < d .(2) The slice rank s k ( P ) is the minimal number r such that P can be writtenin the form P = P ri =1 Q i R i , where deg( Q i ) = 1.(3) In the case when k = F q is a finite field and ψ : k → C ⋆ a non-trivial ad-ditive character we write A k ,ψ ( P ) := P v ∈ V ψ ( P ( v )) q dim( V ) . We define the analyticrank a k ,ψ ( P ) := − log q ( | A F q ,ψ ( P ) | )These three notions of complexity of polynomials play an important role inmany problems in number theory, additive combinatorics, and algebraic geometry.The Schmidt rank, also called the h -invariant, was first introduced by Schmidtin his paper on integer points in varieties defined over the rationals. In his paperSchmidt showed that over the complex field the Schmidt rank of a polynomialsis proportional to the codimension of the singular locus of the associated variety.The same notion of complexity was reintroduced in the work of Ananyan and K. A. is supported by ERC grant 716424 - CASe and ISF Grant 1050/16. T. Z. is supportedby ERC grant ErgComNum 682150 and ISF grant 2112/20.
Hochster [1] as the strength of a polynomial and was used in the proof of theStillman conjecture. The notion of slice rank played an important role in thearguments for the cap set problem in combinatorics [3, 6, 15]. Finally the notionof analytic rank for polynomials over finite fields was introduced in Gower andWolf [10] as a tool for studying the CS-complexity of systems of linear equations.
Remark 1.2.
It is clear that r k ( P ) ≤ s k ( P ) and r k ( P ) = s k ( P ) if d ≤ . In this paper we consider only the case of multilinear polynomials. Namely V = V × · · · × V d is a product of k -vector spaces and P : V × · · · × V d → k is amultilinear polynomial.We introduce two additional definitions. Definition 1.3. (1) In the case when V = V × · · · × V d is a product of k -vector spaces and P : V × · · · × V d → k is a multilinear polynomialwe define a k -subvariety Z P ⊂ V × · · · × V d by Z P := { ( v , . . . , v d ) ∈ V × · · · × V d | P ( v , v , . . . , v d ) = 0 , ∀ v ∈ V } .(2) For a subspace L ⊂ ( V ⊗ . . . ⊗ V d ) ∨ we define r k ( L ) as the minimum of P di =1 codim( W i ) over the set of subspaces W i ⊂ V i , i = 1 , . . . , m , suchthat l | W ⊗ ... ⊗ W d ≡ l ∈ L . Claim 1.4.
For a multilinear polynomial P : V × · · · × V d → k the slice rank s k ( P ) is equal to the minimum of P di =1 codim V i ( W i ) where W i ⊂ V i , ≤ i ≤ d are subspaces such that P | W ×···× W d ≡
0. It follows that if L is one dimensional L = k P then r k ( L ) = s k ( P ). Lemma 1.5.
In the case when k = F q we have A k ,ψ ( P ) = | Z P ( F q ) | q P di =2 dim( Vi ) .Proof. For ( v , . . . , v n ) ∈ V × . . . × V d we define ψ v ,...,v n : V → C ⋆ by ψ v ,...,v n ( v ) := ψ ( P ( v , v , . . . , v n )) . It is clear that ψ v ,...,v n is an additive character of V which trivial if and onlyif ( v , . . . , v n ) ∈ Z P ( F q ). Therefore P v ∈ V ψ v ,...,v n ( v ) = 0 if v Z P ( F q ) and P v ∈ V ψ v ,...,v n ( v ) = q dim( V ) if v ∈ Z P ( F q ). (cid:3) Corollary 1.6.
The analytic rank of a multilinear polynomial P does not dependon the choice of the non-trivial additive character ψ . We will denote it by a k ( P ) . Fro a field k we denote by ¯ k the algebraic closure. Obviously we have r ¯ k ( P ) ≤ r k ( P ). We conjecture that the reverse inequality is essentially holds as well. Conjecture 1.7 (d) . For any d ≥ κ d > r k ( P ) ≤ κ d r ¯ k ( P ) for any field k and a multilinear k -polynomial P of degree d , where ¯ k isthe algebraic closure of k . Remark 1.8.
It is easy to see that Conjecture 1.7 holds for d = 2 with κ = 1. OMPARISON OF THE SCHMIDT AND ANALYTIC RANKS FOR TRILINEAR FORMS 3
In [4] Derksen introduced the notion of the G -stable rank of a multilinearpolynomial (denoted r G k ( P )) defined in terms of Geometric Invariant Theory andproved the following(1) r G k ( P ) = r G ¯ k ( P ) and(2) s k ( P ) d ≤ r G k ( P ) ≤ s k ( P )This immediately implies the inequality s k ( P ) ≤ s ¯ k ( P ) for trilinear polynomials P . Since s k ( P ) = r k ( P ) for trilinear polynomials P we obtain the following result. Theorem 1.9 (Dersken) . For trilinear polynomials we have r k ( P ) ≤ r ¯ k ( P ) . Quantitative estimates for the relation between analytic rank and Schmidtrank provide a means for obtaining quantitative bounds for important problemsin additive combinatorics number theory and algebraic geometry.We conjecture the following quantitative relation between the Schmidt rankand analytic rank.
Conjecture 1.10 (d) . For any d ≥ c d > r F q ( P ) ≤ c d a F q ( P ) for any multilinear polynomial of degree d . Remark 1.11. (1) The inequality a F q ( P ) ≤ r F q ( P ) is known. See [8, 11].(2) It is easy to see that Conjecture 1.10 holds for d = 2 with c = 1.(3) In [13] it was shown that there exist constants c d , e d such that for anymultilinear polynomial of degree d we have r F q ( P ) ≤ c d ( a F q ( P )) e d (earlierwork [2] gave ineffective bounds). Conjecture 1.10 states that one cantake e d = 1.In his paper [14], Schmidt proved the following result Theorem 1.12 (Schmidt) . For any d ≥ there exists D d such that for anycomplex multilinear polynomial P : V × · · · × V d → C we have r C ( P ) ≤ D d g where g := codim V ×···× V d Z P . The main goal of this paper is the proof of the following result.
Theorem 1.13. (1)
Assuming the validity Theorem 1.12 for multilinear poly-nomials of degree d over algebraically closed fields of finite characteristicwe show that Conjecture 1.7(d) implies the validity of Conjecture 1.10(d)for k = F q , q > d with c d = D d κ d − log q ( d ) . (2) Theorem 1.12 holds for multilinear polynomials of degree over alge-braically closed fields of characteristic ≥ with D = 2 . The following statement follows now from Theorem 1.9(2).
Corollary 1.14.
Conjecture 1.10(3) holds with c = 3 . The proof of Theorem 1.13 is based on an extension of the rough bound from[5], and as such the method of proof provided for Conjecture 1.10(3) is completelydifferent that the approach taken in [2], [13].
KARIM ADIPRASITO, DAVID KAZHDAN AND TAMAR ZIEGLER
The conjectured bound in 1.10 can be viewed as a special case of the conjecturedbounded for the inverse theorem for the Gowers norms over finite fields. Let f : V → C . The Gowers uniformity norms of f , introduced in the study ofarithmetic progressions in subsets of the intergers, are defined as follows k f k d U d = 1 | V | d +1 X x,,h ,...,h d ∈ V ∆ h d . . . ∆ h f ( x )where ∆ h f ( x ) = f ( x + h ) f ( x ). Conjecture 1.15.
Let k = F p , and let d > p . There exists a constant F = F ( d, k ) such that the following holds: for any δ >
0, any k -vector space V , any f : V → C satisfying k f k U d ≥ δ , there exists a degree d − P suchthat | | V | P x ∈ V f ( x ) ψ ( P ( x )) | ≥ δ F .When f = ψ ( Q ) where Q is a polynomial of degree d , then ∆ h d . . . ∆ h f ( x ) = ψ ( ˜ Q ( h , . . . , h d )), where ˜ Q is the multilinear symmetric form associated with Q .Via Fourier analysis the conjectured bound in 1.10 implies the validity of Con-jecture 1 .
15 in this case.Finally we present a conjecture relating the rank of a subspace over a field k to its rank over the algebraic closure, and prove it in a special case. Conjecture 1.16.
There exists a constant E d such that r k ( L ) ≤ E d r ¯ k ( L ) forany field k and a linear subspace L ⊂ ( V ⊗ . . . ⊗ V d ) ∨ . Theorem 1.17.
Conjecture 1.16 holds for d = 2 , with E d = 2 . Proof of Theorem 1.17
In this section we prove Theorem 1.17. We start the proof of with the followingresult.
Proposition 2.1.
For any field k and a linear subspace L ⊂ Hom k ( W, V ) thereexists k -subspaces W ′ ⊂ W and V ′ ⊂ V such that dim( W/W ′ ) = dim( V ′ ) =˜ r k ( L ) and l ( W ′ ) ⊂ V ′ for all l ∈ L where ˜ r ( L ) := max B ∈ L r ( B ) .Proof. We start with the follows result of Kronecker (see [9]).
Definition 2.2. (1) Let
V, W be k -vector spaces. A pair A, B ∈ Hom(
V, W )is irreducible is there is no non-trivial decomposition V = V ⊕ V , W = W ⊕ W such A ( V i ) , B ( V i ) ⊂ W i , for i = 1 , n ≥ A n , B n : k n → k n +1 maps such that A n ( e i ) = f i , B n ( e i ) = f i +1 for 1 ≤ i ≤ n , and denote by A ′ n , B ′ n : k n +1 → k n maps such that A ′ n ( e i ) = f i for 1 ≤ i ≤ n , A n ( e n +1 ) = 0 and B ′ n ( e ) =0 , B ′ n ( e i +1 ) = f i for 2 ≤ i ≤ n + 1. OMPARISON OF THE SCHMIDT AND ANALYTIC RANKS FOR TRILINEAR FORMS 5
Claim 2.3.
Let
A, B ∈ Hom(
V, W ) be an irreducible pair of linear maps betweenare finite dimensional k -vector spaces. Then there exist automorphisms T ∈ Aut( V ) , S ∈ Aut( W ) such that either(1) A is an isomorphism or(2) B is an isomorphism or(3) there exist automorphisms T ∈ Aut( V ), S ∈ Aut( W ) such that ( SAT, SBT ) =( A n , B n ) , n = dim( V ) such that ( SAT, SBT ) = ( A n , B n ) or(4) there exist automorphisms T ∈ Aut( V ), S ∈ Aut( W ) such that ( SAT, SBT ) =( A ′ n , B ′ n ) , n = dim( W ). Lemma 2.4. If | k | = ∞ and A, B ∈ Hom k ( V, W ) linear maps such that r ( A ) ≥ r ( A + tB ) for all t ∈ k . Then B (Ker( A )) ⊂ Im( A ) . Remark 2.5.
The assumption that | k | = ∞ is necessary. Indeed in the casewhen k = F q we can take V = W = k [ F q ] with the basis { e x } x ∈ F q , A ( e x ) = xe x and B = Id . Then r ( A + tB ) = r ( A ) = q − t ∈ k but B (Ker( A )) Im( A ). Proof.
Consider first the case when a pair (
A, B ) is irreducible. If Im( A ) = W or Ker( A ) = { } we obviously have the inclusion B (Ker( A )) ⊂ Im( A ). In cases(1) and (4) the map A is onto and in the case (3) the map A is an imbedding.In the case (2) we have dim( V ) = dim( W ) and either A is onto or r ( A ) < r ( B ).We claim that the assumption that r ( A ) < r ( B ) leads to a contradiction.Let S ⊂ A be the subset of s such that r ( s A + B ) < max s ∈ ¯ k r ( sA + B ).Then S ( k ) ⊂ k is finite. So r ( sA + B ) ≥ r ( B ) outside a finite set of s , and since | k | = ∞ and there exists s ∈ k such that r ( A + s − B ) ≥ r ( B ) > r ( A ), which isa contradiction.Now consider the general case when a pair ( A, B ) is a finite direct sum ofirreducible pairs ( A i , B i ) , i ∈ I . Since, as we have seen in the previous paragraph,the condition r ( A i ) ≤ r ( A i + tB i ) is automatically true outside a finite set of t ∈ k and | k | = ∞ the assumption that r ( A ) ≥ r ( A + tB ) for all t ∈ k implies thatthat r ( A i ) ≥ r ( A i + tB i ) outside a finite set of t ∈ k , i ∈ I . So B i (Ker( A i )) ⊂ Im( A i ) , i ∈ I . Therefore B (Ker( A )) ⊂ Im( A ). (cid:3) Lemma 2.6.
Proposition 2.1 holds in the case when | k | = ∞ .Proof. Choose A ∈ L such that r ( A ) = ˜ r ( L ) and define W ′ := Ker( A ) , V ′ :=Im( A ). As follows from Lemma 2.4 we have l ( W ′ ) ⊂ V ′ for all l ∈ L . Butdim( W/W ′ ) = dim( V ′ ) = r ( A ). (cid:3) Now consider the case when the field k is finite. Let G r be the Grassmanianof subspaces W ′ ⊂ W of codimension ˜ r ( L ), and let G r ′ be the Grassmanian ofsubspaces V ′ ⊂ V of dimension ˜ r ( L ) and X ˜ r K ( L ) ⊂ G r × G r ′ be the subvarietyof pairs ( W ′ , V ′ ) such that and l ( W ′ ) ⊂ V ′ for all l ∈ L . It is clear that thesubvariety X ⊂ G r × G r ′ is closed. Therefore it is proper. KARIM ADIPRASITO, DAVID KAZHDAN AND TAMAR ZIEGLER
Let K = k ( t ). Since the field K is infinite it follows from Claim 2.6 that X ( K ) = ∅ . So there exists a rational k -morphism ˆ f : P → X . Any suchmorphism is regular outside a finite k - subset S ⊂ P . So we obtain a regular k -morphism ˜ f : P \ S → X . Since X is proper, ˜ f extends to a regular k - morphism f : P → X . Let ( W ′ , V ′ ) := f (0) ∈ X ( k ). By definition of the variety X we seethat l ( W ′ ) ⊂ V ′ for all l ∈ L . (cid:3) Now we can finish the proof of Theorem 1.17.As follows from Lemma 1.17 there exists k -subspaces V ′ ⊂ V , W ′ ⊂ W suchthat l ( W ′ ) ⊂ V ′ for all l ∈ L ⊗ k K and dim( W/W ′ ) + dim( V ′ ) = 2 r ¯ k ( L ). So r k ( L ) ≤ r ¯ k ( L ). 3. Rough bound
A lemma on the codimension.
Let K be an infinite field, let A be afinitely generated K -algebra of (Krull) dimension n and let L ⊂ A be linearsubspace of A . We denote by J ⊂ A the ideal generated by L and denote by A L the quotient algebra A/J . Lemma 3.1. If dim A L < n then there exists a finite collection of subspaces L i ( L , i ∈ I such that the algebra A l has dimension < n for any l ∈ L \ S i ∈ I L i .Proof. Let X i , i = 1 , . . . , k , be irreducible components of SpecA of dimension n .For every i we denote by K i the field of rational functions on the component X i , and consider the natural morphism ν i : A → K i .Since dim A L < n the image ν i ( L ) is not zero. Hence the subspace L i := L T Ker( ν i ) ⊂ L is strictly contained in L .If an element l ∈ L does not belong to any of the spaces L i then its imagein every field K i is not zero. This implies that the dimension of the algebra A l = A/Al is less than n . (cid:3) Corollary 3.2.
There exist elements l , . . . , l m ∈ L, m := dim( A ) − dim( A/L ) such that dim( A/J ) = dim(
A/L ) where J ⊂ A is the ideal generated by l , . . . , l m .Proof. Induction in m. (cid:3)
A proof of the rough bound.
In this subsection we present a proof of ageneralization of the rough bound from [5].For any subset Θ ⊂ F q [ x , . . . , x n ] we denote by X Θ ⊂ A n the subscheme whichis the intersection of zeros of θ ∈ Θ. Proposition 3.3.
Let M ⊂ F q [ x , . . . , x n ] be a linear subspace of polynomials ofdegrees d such that Y := X M is of dimension m . Then | Y ( F q ) | ≤ q m d c , c := n − m .Proof. Let F be the algebraic closure of F q . OMPARISON OF THE SCHMIDT AND ANALYTIC RANKS FOR TRILINEAR FORMS 7
As follows from Lemma 3.1 there exists P i ∈ M ⊗ F q F, ≤ i ≤ c such thatdim( Y ′ ) = m where Y ′ = X P , P := { P i } ≤ i ≤ c .It is clear that Y ( F q ) is the intersection of Y with hypersurfaces S j , ≤ j ≤ n defined by the equations h j ( x , . . . , x n ) = 0 where h j ( x , . . . , x n ) = x qj − x j .Since Y ⊂ Y ′ we see that Y ( F q ) is contained in the intersection of Y ′ withhypersurfaces S j , ≤ j ≤ n .For j = 1 , . . . , m let H j = P ni =1 a ij h i , a ij ∈ F ′ be a linear combination of the h j such that coefficients a ij are algebraically independent over F where F ′ /F is atranscendental extension. We denote by Z , . . . , Z m ⊂ A n be the correspondinghypersurfaces and define B j := Y ′ ∩ ( T ji =1 Z i ). Claim 3.4.
Each component C of B j is of dimension m − j . Proof.
The proof is by induction in j . The statement obviously is true for j = 0.Any component C of B j +1 is a component of an intersection C ′ ∩ Z j +1 for somecomponent C ′ of B j . By induction dim( C ′ ) = m − j . So not all the functions h j vanish on C ′ . Hence by the genericity of the choice of linear combinations { H j } we see that H j +1 does not vanish on C ′ and therefore C ′ ∩ Z j +1 is of puredimension m − j − (cid:3) As follows from Claim 3.4 the intersection Y ′ ∩ Z ∩ · · · ∩ Z m has dimension 0.Therefore the B´ezout’s theorem implies that | Y ′ ∩ Z ∩ . . . Z n − c | ≤ q m d c . Since Y ( F q ) = Y ∩ Z ∩ · · · ∩ Z n ⊂ X ∩ Y ∩ · · · ∩ Y n − c we see that | Y ( F q ) | ≤ q m d c . (cid:3) Proof of Theorem 1.13(1)
Proof.
Let k = F q , P : V × · · · × V d → k be a multilinear polynomial and g = codim V ×···× V d Z P and ¯ k be the algebraic closure of F q .Since ( see Lemma 1.5) a k ( P ) = P di =2 dim( V i ) − log q ( | Z P ( k ) | ) it follows fromProposition 3.3 that | Z P ( k ) | ≤ d g q n − g where n = P di =2 dim( V i ). So log q ( | Z P ( k ) | ) ≤ g log q ( d ) + n − g . We see that a k ( P ) ≥ g (1 − log q ( d )).Assuming the validity of Theorem 1.12 for multilinear polynomials of degree d over ¯ k we see that g ≥ r k ( P ) κ d D d and therefore a ( P ) ≥ r k ( P ) κ d D d (1 − log q ( d )). Theorem1.13(1) is proven. (cid:3) In the next section we prove the second part of Theorem 1.13.5.
The adaptation of the Schmidt’s result for fields of finitecharacteristic in the case when d = 3 . We use notations from Definitions 1.1,1.3. In [14] W. Schmidt proved thefollowing result.
Theorem 5.1 (Schmidt) . For any d ≥ there exists D d such that for anycomplex multilinear polynomial P : V × · · · × V d → C we have r C ( P ) ≤ D d g where g := codim V ×···× V d Z P KARIM ADIPRASITO, DAVID KAZHDAN AND TAMAR ZIEGLER
His proof extends to any algebraically closed field, and we sketch it here forthe three-dimensional case.In this section we fix an algebraically closed field k and write r ( P ) instead of r k ( P ). Let X be an algebraic k -variety. Since our field k is algebraically closedany constructible subset Y ⊂ X defines uniquely a subset Y ⊂ X such that Y = Y ( k ). We say that a constructible subset Y ⊂ X is open if the subset Y ⊂ X is open and we define dim( Y ) := dim( Y ).For a trilinear polynomial P : U × V × W → k we denote by Z P ⊂ V × W theconstructible subset of points ( v, w ) such that P ( u, v, w ) = 0 for all u ∈ U .The main goal of this section is to prove the following theorem: Theorem 5.2.
For any trilinear polynomial P : U × V × W → k we have r ( P ) ≤ g where g := codim V × W Z P . Before proving Theorem 5.2 we remind the reader of some results from Algebra.5.1.
Linear Algebra.Definition 5.3.
For a bilinear form B : U × V → k we write(1) S U ( B ) := { u ∈ U | B ( u, v ) = 0 , ∀ v ∈ V } .(2) S V ( B ) := { v ∈ V | B ( u, v ) = 0 , ∀ u ∈ U } . Remark 5.4. r ( B ) = codim U S U ( B ) = codim V S V ( B ).Let X be a smooth curve over k , let t ∈ X and let B ( t ) : U × V → k , t ∈ X be a family of bilinear forms. We write B := B ( t ), S U := S U ( B ) , S V := S V ( B ),and C := ( ∂B ( t ) /∂t ) t = t . Claim 5.5. If r ( B ( t )) ≤ r ( B ) , t ∈ X then C | S U × S V ≡ Proof.
We show that the assumption that C | S U × S V X = A and t = 0.To start with we choose bases e i , f j , where 1 ≤ i ≤ dim( U ) , ≤ j ≤ dim( V )of U and V such that(1) B ( e i , f j ) = δ i,j , ≤ i, j ≤ r ( B ).(2) B ( e i , f j ) = 0 if either i > r ( B ) or j > r ( B ).Then S U is the span of { e i } , where r ( B ) + 1 ≤ i ≤ dim( U ) and S V is the spanof { f j } , where r ( B ) + 1 ≤ j ≤ dim( V ). Since C | S U × S V i , j ) such that i , j > r ( B ) and C ( e i , f j ) = 0.Let U ′ ⊂ U , dim( U ′ ) = r ( B ) + 1 be the span of { e i } ≤ i ≤ r ( B ) and of e i and V ′ ⊂ V , dim( V ′ ) = r ( B ) + 1 be the span of { f j } , ≤ j ≤ r ( B ) and of f j .We denote by ∆( t ) be the determinant of the bilinear form of B ( t ) U ′ × V ′ withrespect to chosen bases of U ′ and V ′ . The condition r ( B ( t )) ≤ r ( B ) , t ∈ k impliesthat ∆( t ) ≡
0. On the other hand ∆( t ) ≡ tC ( e i , f j ) mod ( t ).This contradiction proves Lemma 5.5. (cid:3) OMPARISON OF THE SCHMIDT AND ANALYTIC RANKS FOR TRILINEAR FORMS 9
Transcendence degree.Definition 5.6. (1) For a field extension K/ k we define the transcendencedegree tr ( K/ k ) to be the minimal n ≥ k ( u , u , . . . , u n ) ֒ → K .(2) For a point w ∈ K N we denote by k ( w ) ⊂ K the subfield generated by k and the coordinates of w and write tr k ( w ) := tr ( k ( w ) / k ).(3) Let X ⊂ A N be an irreducible k -subvariety. A point w ∈ X ( K ) is genericif it is not contained in any proper k -subvariety of X . Claim 5.7. (1) Let X ⊂ A M be an irreducible k variety and w ∈ X ( K ) begeneric point. Then tr ( k ( w ) / k ) = dim( X ).(2) Let p : A M → A N be a k -linear map, Z ⊂ A M an irreducible variety,and Y = p ( Z ). Let v ∈ Z ( K ) be generic point and w := p ( v ) ∈ Y ( K ).Then w ∈ Y ( K ) is a generic point, k ( w ) ⊂ k ( v ) and tr ( k ( v ) / k ( w )) =dim( Z ) − dim( Y ).5.3. Proof of Theorem 5.2.
Proof.
Let P : U × V × W → k be as in Theorem 5.2 and Z ⊂ Z P be an irreduciblecomponent of the maximal dimension, g := codim V × W Z . We have to show that r ( P ) ≤ g .Let z = ( v , w ) be a generic point of Z , and let Y ⊂ W be the projectionof Z on W . For w ∈ W we denote by P w the bilinear forms on V given by P w ( u, v ) := P ( u, v, w ).Let S U = { u ∈ U | P w ( u, v ) = 0 , ∀ v ∈ V } , and let S V = { v ∈ V | P w ( u, v ) =0 , ∀ u ∈ U } . Since w is a generic point of Y , we see that r ( P y ) ≤ r ( P w ) for all y ∈ Y . Since w ∈ Y is a generic point, Y is smooth at w and we can define S W := T Y ( w ) the tangent space at w ..It follows from Calim 1.4 that the following statement implies the validity ofTheorem 5.2. Proposition 5.8. (1) P | S U × S V × S W ≡ . (2) codim U ( S U ) + codim W ( S W ) + codim V ( S V ) ≤ g .Proof. To show that P | S U × S V × S W ≡ C ∈ S W therestriction of the bilinear form P C on S U × S V is equal to 0. So we fix C ∈ S W .Since P w for w ∈ W is a linear function on W we have P C = ( ∂P w /∂C ) w = w . Sowe have to show that ( ∂P w /∂C ) w = w | S U × S V ≡ Y passing through w and tangent to C . In otherwords choose a map φ : X → Y of a smooth curve X to Y and a point t ∈ X suchsuch that φ ( t ) = w and C = ( ∂φ ( t ) /∂t ) t = t . Since r ( P y ) ≤ r ( P w ) for y ∈ Y thefamily B ( t ) := P φ ( t ) of bilinear forms on U × V satisfies the assumption of Claim5.5. Therefore ( ∂P w /∂C ) w = w | S U × S V ≡
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