| The study of polynomial automorphisms is very important to the research of algebra and geometry. Jacobian Conjecture is a celebrated problem on polynomial automor-phisms, and has attracted a lot of attentions since it was posed. During the research of this conjecture, people keep on giving new ideas, new questions, and new methods. This does not only promote the integration of related fields, but also promote the study of polynomial automorphisms.Let K be a field, X=(X1;…,Xn), and K[X] the polynomial ring on K. Assume that Fi E K[X]. Then F=(F1,…,Fm):Kn→Km defines a polynomial map, such that The composition of polynomial maps F:Kn→Km and G:Km→Kr is defined by Furthermore, if GoF=FoG=X, then F is an automorphism over Kn or invertible.Jacobian Conjecture JC(K,n) Let K be a field of characteristic0. Suppose a polynomial map F=(F1;…, Fn) satisfies det JF E K*. Then F is invertible.Although a lot of people have done excellent work to attack Jacobian Conjecture, it is still open till now. Most of the results essentially fall into the following categories, the reduction of the conjecture and special polynomial maps. In Chapter1, we give a systematical survey about Jacobian Conjecture.In Chapter2, using the homogenization method and Gorni-Zamperi pairing, we establish the connection between general polynomial maps and polynomial maps of the form F=X+(AX)(d). Then we give a sufficient and necessary condition for a polynomial map to be an injective map. Finally, we get a sufficient and necessary condition for a polynomial map to be a polynomial automorphism.We show that there exist ri∈C, such that (x+y)d—yd=d/d-1∑i-1d-1(rix+y)d-1x.Theorem2.2.2A polynomial map F with degF <4is invertible if and only if for all α, β∈Cn.Theorem2.2.3A polynomial map F of degree d is invertible if and only if for all β,γ∈CnWe show that the syzygy bases of X—Y is the set of sys(X—Y)={ri,j|l≤i <j <m} with ri,j=(0,...,0, Yj-Xj,0,...0, Xi-Yi,0,...,0). Let SYS(X-Y) be the module generated by sys(X—Y). There exists a matrix M(X,Y) such that F(X)-F(Y)=M(X,Y)(X-Y). Given a matrix A, denote by Ai the i-th row of matrix A.Theorem2.4.2A polynomial map F is a polynomial automorphism if and only if there exists a polynomial matrix N(X, Y) such that for all1<i <n.Finally, we give an equivalent conjecture of Jacobian Conjecture at the end of the chapter:Conjecture2.5Assume a polynomial map F satisfies det JF∈C*. Then there exists polynomial matrixes M(X,Y) and N(X,Y) such that(1) F(X)-F{Y)=M(X, Y)(X-Y);(2)(N(X, Y)M(X, Y)-I)i∈SYS(X-Y). In Chapter3, we study polynomial maps F with∑i=1m JF(αi) invertible, by studying the directional derivatives along lines.Theorem3.1.6Let F: Cn→Cm be a polynomial map of degree <d and β,γ∈Cn Then the following statements are equivalent.(1) There exists λi G C satisfying∑i∈Iλi≠0for all nonempty I (?){1,2,...,d—1}, such that(2) F|β+cγ is linearly rectifiable (in particular injective), i.e. there exists a vector v G Cn such that(3) For all s G N, for all λi G C such that λ1+λ2+…λs≠0.Theorem3.3.9For a polynomial map F: Cn→Cn the following statements are equivalent.(1)∑in=1(JF)|αi is invertible for all αi∈Cn(2) F=L o (X+H), where H has no linear terms, the linear part L of F is invertible and JH is additive-nilpotent;(3) F=(X+H) o L, where H has no linear terms, the linear part L of F is invertible and JH is additive-nilpotent;(4) F=L|o (X+H) o L2, where L1and L2are invertible maps of degree one and JH is additive-nilpotent.We end the chapter by giving a numerical method to check if∑i=1m JF(αi)is invert-ible for a polynomial map F.In Chapter4we study the structure of polynomial automorphisms of the form F X+(AX)(3): C6→C6.Theorem4.1.1Let F=X+(AX)(3): C6→C6be a polynomial automorphism. Then there exists T G Gl6(C) such that T-1FT=X+(5X)(3), with B an upper triangular matrix with zero diagonal. Finally, using Maple procedure we determine relations of entries of B.
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