| The Jacobian Conjecture is a famous open problem in affine algebra geometry. Let k be a field of characteristic zero and F : kn→kn a polynomial map. The Jacobian Conjecture asserts that F is invertible if the Jacobian determinant of F is a nonzero constant. The conjecture was formulated by Keller in 1939. During the last seventy years, many mathematicians are interested in the study of the Jacobian Conjecture, and many achievements have been obtained. In 1982, Bass et al. proved that for the Jacobian Conjecture, it suffices to investigate the so-called Yagzhev maps, i.e., polynomial maps of the form X + H with JH nilpotent, where H is cubic homogeneous. The studies of nilpotent Jacobian matrices led to the Homogeneous Dependence Problem (HDP(n, d)): Let H be a homogeneous polynomial map of degree d≥1, such that JH is nilpotent. Dose it follow that the components of H are linearly dependent over k; equivalently. are the rows of JH linearly dependent over k?This dissertation is devoted to three topics on polynomial maps: the linear dependence problem for power linear maps, the linear dependence problem for Yagzhev quasi-translations in dimension 5 and the strongly nilpotent problem for quadratic Yagzhev maps.A polynomial map of the form H = (?) is called a power linear map of degree d. H can be written as (AX)(d), where A = (aij)n×n. A polynomial map of the form F = X + (AX)(d) is called a Druzkowski map of degree d. In 1983, Druzkowski showed that it is sufficient to consider in the Jacobian Conjecture for every n > 1 only for the cubic Druzkowski maps. In 1995, Olech formulated the homogeneous dependence problem for cubic linear maps when he studied the Markus-Yamabe conjecture. In this dissertation, we study the homogeneous dependence problem for power linear maps of degree d, which is denoted by PLDP(n, d). First we settle a conjecture, due to Gorni and Tutaj-Gasinska, about an invariant of the Druzkowski maps. Second by use of the result, we give a necessary and sufficient condition for the components of H = (AX)(d) to be linearly dependent, and the relations between HDP and PLDP are formulated. Finally, an algorithm (including a Mathematica routine) is given to compute the counterexamples to PLDP(n, d), and the counterexamples are given for the cases n≥67, d = 3 and n≥48, d = 4.Let F = X + H be a polynomial map. Then F is called a quasi-translation if F is invertible and F-1 = X - H, and F is called a Yagzhev quasi-translation if additionally F is a Yagzhev map. In 1876, P. Gordan and M. Nother studied the quasi-translations in order to understand singular Hesse matrices better. They showed that the components of H are linearly dependent if F = X + H is a Yagzhev quasi-translation in dimension 4. In 2004, de Bondt found the counterexamples to HDP(n, 5) in the Yagzhev quasi-translations for all n > 6. However the following problem is still open: Let F = X + H be a Yagzhev quasi-translation in dimension 5, dose it follow that the components of H are linearly dependent? Quasi-translations can also be seen as a special type of locallynilpotent derivation——the nice derivations. Let D be a derivation on k[X]. Then Dis called a nice derivation if D2(Xi) = 0,1≤i≤n. For a polynomial map F = X + H, let D =(?). Then, F = X + H is a (Yagzhev) quasi-translation if and only if D is a (homogeneous) nice derivation, and the components of H are linearly dependent if and only if the kernel of D contains a coordinate. In this dissertation, we study the homogeneous problem for the Yagzhev quasi-translation F = X + H of degree d in dimension 5, we show that if d≤11, then the components of H are linearly dependent, and as a corollary we show that the kernel of a homogeneous nice derivation of degree d≤11 in dimension 5 contains a coordinate.Let H be a polynomial map in dimension n over a field k of characteristic zero. The Jacobian matrix JH is called strongly nilpotent if JH(α1)JH(α2)…JH(αn) = 0 for allα1,α2,…,α2∈kn. A polynomial map F = X + H is called a triangular automorphism, if Hi∈k[Xi+1,…, Xn] for all 1≤i≤n - 1 and Hn∈k. And F is called 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 1991, Meisters and Olech showed the following result: Let F = X + H be a quadratic Yagzhev map with JH nilpotent. Then JH is strongly nilpotent if n≤4 or JH2 = 0, and JH is not necessarily strongly nilpotent if n≥5 and JH4 = 0. And they formulated the following problem: Let F = X + H be a quadratic Yagzhev map with JH3 = 0, does it imply that JH is strongly nilpotent? In this dissertation, we show that the problem has an affirmative answer in dimension n≤6.The main results of this dissertation are as follows.Theorem 2.1.1 For any n×n positive semidefinite complex matrix B and any positive integer d,Theorem 2.1.2 For any n×n complex matrix A and any positive integer d,Theorem 2.2.1 Let H = (AX)(d): Cn→Cn. Then the components of H have the same linear dependence as the rows of matrix (AA*)(d) In particular, the components of H are linear dependent if and only if det(AA*)(d) = 0.Theorem 2.2.2 If the components of (AX)(d0) are linearly independent, then the components of (AX)(d) are linearly independent for any integer d≥d0. In particular, if the rows of A are linearly independent, then for any positive integer d, the components of (AX)(d) are linearly independent.Theorem 2.2.3 Let f= X + H : Cn→Cn and FA = Y + (AY)(d) : CN→CN be a Gorni-Zampieri pair. Then the components of H are linearly independent if and only if rankA(AA*)(d) < rankA.Theorem 2.2.4 Let f= X + H : Cn0→Cn0 and FA = Y + (AY)(d) : CN→CN be a Gorni-Zampieri pair. If H is a counterexample to HDP(n0, d) with rank(AA*)(d) = n1, then there exist counterexamples to PLDP(n,d) for all integers n≥n1.Corollary 2.2.1 For a fixed integer d≥2, PLDP(n, d) has an affirmative answer for all positive integers n if and only if HDP(n, d) has an affirmative answer for all positive integers n.Let F = X + H be a Yagzhev quasi-translation in dimension 5 over C of the formX + (h1 (p, q), h2(p, q), h3(p, q), h4(p, q), r) (*)such that p,q∈C[X] are homogeneous with deg p = deg q = t, h1,h2,h3,h4∈C[U, V] are homogeneous with deg hi = s, 1≤i≤4, p, q are irreducible, p, q, r are algebraiclly independent, and the components of H are relatively prime. Theorem 3.2.1 Let F = X + H be a Yagzhev quasi-translation in dimension 5 of the form s (*). If s≤2, then the components of H are linearly dependent.Theorem 3.2.2 L (?) = X + H be a Yagzhev quasi-translation in dimension 5 of the form as (*). If t≤3, then the components of H are linearly dependent.Theorem 3.2.3 Let F = X + H be a Yagzhev quasi-translation of degree d in dimension 5. If d≤11, then the components of H are linearly dependent.Corollary 3.2.2 Let D be a homogeneous nice derivation of degree d in dimension 5 . If d≤11, then rank D < 5, i.e., ker D contains a coordinate.Theorem 4.2.2 Let F = X + H be a quadratic Yagzhev map with JH3 = 0. If rankJH≤3, then the components of H are linearly dependent.Corollary 4.2.1 Let F = X + H be a quadratic Yagzhev map with JH3 =0 in dimension n≤5. Then JH is strongly nilpotent.Theorem 4.3.1 Let F = X + H be a quadratic Yagzhev map with JH3=0 indimension 6. Then JH is strongly nilpotent.
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