Polynomial Maps With Additive-nilpotent Jacobian Matrix

Several important problems in affine algebraic geometry, such as the Jacobian Conjecture and the Tame Generators Problem, are closely related to the research of polynomial automorphisms and derivations on polynomial rings, the study of which is growing rapidly in the last several decades. Let k be a field of characteristic zero. The Jacobian Conjecture asserts that a polynomial map over k is invertible if the Jacobian determinant of which is a nonzero constant. In the 1980s, a famous reduction result to the Jacobian Conjecture was obtained by Bass et al.,which asserts that it suffices to verify the Jacobian Conjecture for all the polynomial maps of the form X + H,where J H is nilpotent, and one can even assume that H is cubic homogeneous, i.e., each H_i is either zero or homogeneous of degree 3. Since then many authors are concerned with the polynomial maps with nilpotent Jacobian matrix and many results were obtained.In this dissertation, we mainly focus on the polynomial maps with so-called "additive-nilpotent" Jacobian matrix. First we prove that the Jacobian Conjecture holds for these maps and then the structure of these maps are discussed, especially the structure of the quadratic homogeneous polynomial automorphisms.In 1991, Meisters and Olech introduced the concept of strongly nilpotent Jacobian matrix, which was generalized to the following form by van den Essen and Hubbers in 1996: for a polynomial map H of dimension n over k,J H is called strongly nilpotent if JH(α₁)J H(α₂)…J H(α_n) = 0 for all vectorsα₁,α₂,…,α_n∈kⁿ.Let F = X + H be a polynomial map, we say that F is a triangular automorphism if H_i∈k[X_i+1,…,X_n] for all 1≤i≤n-1 and H_n∈k.We say that F is linearly triangularizable if F is linearly conjugate to a triangular automorphism. Van den Essen and Hubbers showed that for a polynomial map F = X + H,JH is strongly nilpotent if and only if F is linearly triangularizable. In this dissertation, we introduce a new class of nilpotent Jacobian matrix, which is called "additive-nilpotent".For a polynomial map H of dimension n over k,JH is called additive-nilpotent if the linear subspace spanned by {JH(α)|α∈kⁿ} is a nilpotent subspace, i.e, every element of which is nilpotent. We prove that a polynomial map F=X+H with JH additive-nilpotent is a polynomial automorphism, and the properties of such automorphisms are discussed. In addition, the problem of how to determine if a Jacobian matrix is additive-nilpotent is also discussed and an algorithm is given.It is important to describe the structure of polynomial automorphisms. Let k[X]:= k[X₁,…,X_n] be the polynomial ring in the variables X₁,…,X_n over k. The group of kautomorphisms of k[X] is denoted by Aut_kk[X].A polynomial automorphism of the form E=(X₁,…,X_i+α,…,X_n) is called elementary,where a∈k[X₁,…,(?)_i,…,X_n].A polynomial automorphism with degree 1 is called affine.The subgroup of Aut_kk[λ'] generated by all the elementary automorphisms and all the affine automorphisms is called the tame subgroup, and is denoted by T(k,n). The Tame Generators Problem asks if Aut_kk[λ]=T(k,n),in other words if every polynomial automorphism is tame. It turned out that the answer to this problem is yes for dimension 2 which is known as the Jungvan der Kulk theorem. An automorphismσin dimension 3 was given by Nagata in 1972 and he conjectured thatσis a counterexample to the Tame Generators Problem. The conjecture of Nagata was finally verified by Shestakov and Umirbaev in 2002. However, Rusek conjectured that every quadratic polynomial automorphism is tame. It is suffices to investigate the Rusek Conjecture for all the quadratic homogeneous polynomial automorphisms.The Rusek Conjecture was only verified in some special cases. In 1991, Meisters and Olech proved that, a quadratic homogeneous polynomial automorphism is linearly triangularizable in the following cases: (1)n≤4;(2) J H²=0. The difficulty will growth rapidly when the dimension n or the nilpotency index of J H increases. In this dissertation,we classify all quadratic homogeneous polynomial automorphisms in dimension 5 over C, and we show that they are all tame.In addition, we discuss the structure of quadratic homogeneous quasi-translations, and we show that they are linearly triangularizable when the dimension is no more than 10. In fact, the description of quasi-translation also describe the properties of a special type of locally nilpotent derivation, the so-called nice derivation.The main results of this dissertation are as follows. Theorem 2.2.1 Let F=X + H be a polynomial map. If JH is additive-nilpotent, then F is a polynomial automorphism.In what follows, letand let w(p_d-1) be the width of the homogeneous polynomial p_d-1Theorem 2.2.4 Let F = X + H be a homogeneous polynomial map of degree d. Iffor anyα₁,…,α_w(pd-1) kⁿ,is nilpotent, then F is a polynomial automorphism.Corollary 2.2.2 Let F = X + H be a homogeneous polynomial map of degree d. If for anyα₁,…,α_d-1∈kⁿ,is nilpotent, then F is a polynomial automorphism.Theorem 2.2.5 Let F=X + H be a polynomial map of degree d. If for anyα₁,…,α_w(pd)∈kⁿ,the subspace spanned by J H(α₁),…,J H(α_w(pd)) is nilpotent, then F is a polynomial automorphism.Corollary 2.2.3 Let F = X + H be a polynomial map of degree d. If for anyα₁,…,α_d∈kⁿ,the subspace spanned by J H(α₁),…,J H(α_d) is nilpotent, then F is a polynomial automorphism.Theorem 2.3.2 Let H =(AH)^(d) be a polynomial map over C. Then J H is additive-nilpotent if and only if dia,g((AA^*)^(d-1)X) A is nilpotent.Theorem 2.3.3 Let f = X + H :Cⁿ→Cⁿ and F_A=Y+(AY)^(d):C^N→C^N be a Gorni-Zampieri pair. Thenwhere P=(E₁₁,E₂₂,…,E_nn)^T and E_ii denotes the matrix unit.An algorithm is also given to determine if a Jacobian matrix is additive-nilpotent.Theorem 3.3.2 Let F = X + H be a quadratic homogeneous quasi-translation. If rankJ H≤6,then F is linearly triangularizable. In particular, if n≤10, then F is linearly triangularizable. Corollary 3.3.3 Let D be a quadratic homogeneous nice derivation in dimensionn(≤10).Then D is linearly conjugate to a triangular derivation.Theorem 3.4.2 Let F = X + H be a quadratic homogeneous quasi-translation. If rankJ H≤7,then the components of H are linearly dependent. In particular, if n≤12, then the components of H are linearly dependent.Corollary 3.4.1 Let D be a quadratic homogeneous nice derivation in dimension n(≤12). Then ker D contains a linear coordinate of k[X₁,…,X₁₂].Theorem 4.1.1 HDP(5,2) has an affirmative answer, that is, for any quadratic homogeneous polynomial automorphism F = X + H, the components of H are linearly dependent.Theorem 4.2.2 Let F = X + H be a quadratic homogeneous automorphism in dimension 5 over C.Then F is either linearly triangularizable or linearly conjugate to one of the following two automorphisms.whereÏ„_i∈k[X₄,X₅],i=1,2,3,are quadratic homogeneous polynomials.whereα₂≠0. Furthermore, F' and F" are not conjugate.Theorem 4.2.3 Every quadratic homogeneous automorphism in dimension 5 over C is tame.