The Dependence Problem For Power-linear Maps

The Jacobian Conjecture is a famous open problem in affine algebraicgeometry. It asserts that a polynomial map F = ( F₁ , … , F_n) from n to nis invertible if the Jacobian matrix JF= ( (?)F_i / (?)x_j )1≤ i ,j ≤n is invertible. Bass,Connell and Wright in 1982 showed that it suffices to prove the JacobianConjecture for polynomial maps of the form F = x + H with JH nilpotent. Infact they even showed that one may assume that H is homogeneous of degree3, i.e., each component H_i is either zero or homogeneous of degree 3. Thestudies of the nilpotent Jacobian matrices led to the following problem.Homogeneous Dependence Problem (HDP(n, d)):Let H = ( H₁, … , H+n): Cⁿ →Cⁿ be homogeneous of degree d≥1 such thatJH is nilpotent. Does it follow that H 1, … , Hn are linearly dependent over C(or equivalently, are the rows of JH linearly dependent over C )?Affirmative answers are known in the case n≤3, d arbitrary and the casen = 4, d = 3. Counterexamples are given by de Bondt for all dimensions n≥5,including counterexamples for all dimensions n≥10, d = 3. But HDP(n, 2) isstill an open problem.Druzkowski showed that it suffices to prove the Jacobian Conjecture forpolynomial maps of the form F = x + ( Ax)⁽³⁾. In general, F = x + ( Ax)^d iscalled a Druzkowski map of degree d. The polynomial map H = ( Ax)^(d) iscalled a power-linear map of degree d. In this thesis we are concerned with thefollowing problem.Power-linear Dependence Problem (PLDP(n, d)):Let H = ( Ax) ( d ):n → n, d≥1, such that JH is nilpotent. Does itfollow that H 1, … , Hn are linearly dependent over (or equivalently, arethe rows of JH linearly dependent over )?We give a necessary and sufficient condition for the components ofH = ( Ax) ( d) to be linearly dependent by use of a power-linear similarityinvariant, and established connections between HDP(n, d) and PLDP(n, d).Finally, we study the PLDP(n, 2) and strongly nilpotent Jacobian matrices, andwe prove that if the nilpotency index of JH for H = ( Ax)(2) is less than 4,then JH is strongly nilpotent, and consequently the rows of JH are linearlydependent.We start with some basic definitions.Definition 2.1.2 Let F = x + Hbe polynomial map.(1) We say that F is in triangular form if H i ∈ [ xi +1, … , xn] for all1 ≤ i ≤ n? 1 and H n∈;(2) We say that F is linearly triangularizable if there exists T ∈ Gln( )such that T ?1 FT is in triangular form.Definition 3.1 For A = ( ai j ) ∈ M m× n( ) and B = (b i j ) ∈ M m× n( ), thematrix ( ai j bi j ) ∈ M m× n( ) is called Hadamard product of A and B and isdenoted by A B . A( k ) = ( aikj) is called the kth Hadamard power of A.Definition 3.3 Let FA = x + ( Ax) ( d) and FB = x + ( Bx) ( d) be twoDruzkowski maps of degree d. FA and FB (or A and B) are calledpower-d-similar, denoted bydFA ～ FB (ordA ～ B), if there exists T ∈ Gln( )such that FB = T ?1FA T.The dimension of span ( x ( Ax ) ( d)) is a power-d-similarity invariant.In fact, dim span ( x ( Ax ) ( d ) ) = rank( AA* )( d). Using this invariant we obtainan equivalent condition for dependence of the components of H = ( Ax) ( d).Theorem 3.6 Let H = ( Ax) ( d) be a power-linear map. Then thecomponents of H have the same linear dependence as the rows of matrix( AA* ) ( d). In particular, the components of H are linearly dependent if and onlyif det( AA* ) ( d)= 0.Theorem 3.10 If the components of H = ( Ax) ( d) are linearly dependent,then so is H = ( Ax) ( k) for any integer 1 ≤ k ≤ d? 1.Proposition 3.12 Let H = ( Ax) ( d) with JH nilpotent and rankA = r. IfHDP(r, d) is true, then the components of H are linearly dependent.Since HDP(n, d) has affirmative answer in the case n≤3, d arbitrary andthe case n = 4, d = 3 . We have the following resultCorollary 3.13 Let H = ( Ax) ( d) with JH nilpotent. Then the componentsof H are linearly dependent in the case rankA≤3, d arbitrary and the caserankA = 4, d = 3.Proposition 3.14 Let H = ( Ax ) ( d), d ≥ 3 such that JH is nilpotent andcorank A≤2, then the components of H are linearly dependent.Corollary 3.15 PLDP(n, d) has affirmative answer in the case n≤6, darbitrary and the case n = 7, d = 3 .The dependence and the structure of polynomial maps have closerelations. The following is such a result.Theorem 3.18 Let H = ( Ax ) ( d), d ≥ 3 such that JH is nilpotent. Ifrank ( AA* ) ( d) ≤ rank A + 1, then there exists strictly upper-triangular matrixB such thatdA ～ B, and consequently F = x + His linearly triangularizable.Finally we study the PLDP(n, 2) by use of the notion of strongly nilpotentJacobian matrix. Let Y( 1) = (Y ( 1)1 , … , Y( 1) n ), … , Y( n ) =(Y ( n )1 , … , Y( n )n) be n sets ofn new variables.Definition 4.1 The Jacobian matrix JH is called strongly nilpotent ifJH (Y ( 1) ) JH (Y ( 2) ) JH (Y ( n)) = 0.The condition in Definition 4.1 is equivalent toJH (α 1 ) JH(α n ) = 0, ? α 1, … , αn∈n.Definition 4.7 Let f = x + H: n →nbe a polynomial map, where His homogeneous of degree d≥2 and FA = y + ( Ay) ( d ): N →N withN > n. We say that f and FA are a Gorni-Zampieri pair through the matricesB ∈ Mn , N( ) and C ∈ MN , n( ) if(1) f ( x ) = BFA (C x);(2) BC = In;(3) ker B = ker A.Theorem 4.12 If f and FA are a Gorni-Zampieri pair , then(1) Ind J ( Ay ) ( d) = Ind JH + 1, where Ind means the nilpotency index;(2) JH is strongly nilpotent iff J ( Ay ) ( d) is strongly nilpotent.Equivalently, f = x + H is linearly triangularizable iff FA is linearly triang-ularizable.Theorem 4.13 Let F = x + ( Ax)(2). If ( J ( Ax ) (2) ) 3= 0, then J ( Ax )(2) isstrongly nilpotent, and consequently the rows of J ( Ax )(2) are linearlydependent.Definition 4.14 Let F = x + H be a polynomial map. If F is invertibleand F ?1 = x ? H, then F is called a quasi-translation.de Bondt makes use of the quasi-translation to construct counterexamplesof HDP(n, d). Does exists a quasi-translation to be a counterexample ofPLDP(n, d)? The following proposition gives a negative answer.Proposition 4.16 Let F = x + ( Ax) ( d) is a quasi-translation. Then F islinearly triangularizable, or equivalently, J ( Ax ) ( d) is strongly nilpotent.Moreover, the rows of J ( Ax ) ( d) are linearly dependent.