Co-tame Automorphisms Of Polynomial Algebras

Affine algebraic geometry mainly studies the automorphisms of affine space A_kⁿ,and from the point of view of algebra it is to study the automorphisms of polynomial algebra k[x₁,...,x_n].In this paper,we always assume that k is a field and chark=0.A basic problem in this field is to study the structure and relationship of the automorphisms group GA_n（k）of polynomial algebra k[x₁,...,x_n]and its important subgroups,such as affine subgroup Aff_n（k）,tame subgroup TA_n（k）,triangular automorphisms subgroup BA_n（k）and porobolic subgroup PA_n（k）.In this paper we mainly study co-tame automorphisms.Givenσ∈GA_n（k）we say thatσis co-tame if TA_n（k）?<Aff_n（k）,σ>.Such automorphisms are helpful to understand the structure of GA_n（k）.In this thesis,the first chapter introduces the background of the de-velopment of co-tame automorphisms.The second chapter introduces the research results of co-tame automorphisms in the literature,especially the results on the co-tame automor-phisms in TA_n（k）.The third chapter introduces the co-tame automorphisms that outside the group TA_n（k）and also introduces the stable co-tame automorphisms.In the fourth chapter are our own results on co-tame automorphisms.We mainly obtain the following results.Theorem 0.1 Let n>2,φ=τ₁α₁τ₂α₂τ₃be a non-affine automorphism,whereα_i∈Aff_n（k）（1≤i≤2）,τ_i∈PA_n（k）（i=1,3）,τ₂∈BA_n（k）.Thenφis co-tame.Theorem 0.2 Let F be a non-affine automorphism in dimension 4.If F is additive-nilpotent,then F is co-tame.Theorem 0.3 Let F=x+H be a quadratic homogeneous non-affine automorphism in dimension 5.Then F is co-tame.Theorem 0.4 Let n>2,F=x+H be a quadratic homogeneous non-affine automor-phism in dimension n.If rank JH≤3,then F is co-tame.Theorem 0.5 Let n>2,F=x+H be a cubic homogeneous non-affine automorphism in dimension n.If rank JH≤2,then F is co-tame.Theorem 0.6 Let n>2,F=x+H,H=（Ax）^（d）be a non-affine Dru˙zkowski automorphism in dimension n.Then F is co-tame.