The Linearization Problem Of Polynomial Automorphisms

From the point of view of algebra,afine algebraic geometry mainly studies the au-tomorphisms of polynomial algebra k[x1,…,xn].Let k be an algebraically closed field,(?)is a n-dimensional affine space on k,G is a linearly reductive group,A:Gx V? V is a group action with a fixed point.For g?G,let A(g)denote the corresponding poly-nomial automorphism of V or the corresponding automorphism of k[x1,…,xn].A well-known linearization problem in this field is:is A conjugate to a linear action L:Gx V? V under some conditions,that is,whether there is an automorphism a of k[x1,…,xn],such that ?-1A(g)?=L(g).In this paper,we proved that a class of automorphisms of a poly-nomial algebra is conjugated to its linear part,thus we proved that the corresponding group action can be linearized,and we characterized the polynomial retracts with sparse homogeneous parts.This article mainly characterizes the polynomial map ? under the different conditions,studies the problem that whether it can be linearized under the conditions ?2=I and?2=c?,c?0.It is mainly composed of four parts.The first chapter mainly introduces the development background of the linearization problem and some basic knowledge about polynomial automorphism.The second chapter introduces the basic content of polyno-mial retracts in existing literatures,the existing conclusions of the polynomial retracts with sparse homogeneous parts are emphasised.The third chapter mainly summarizes some results in the existing literature on whether the linearly reductive groups can be linearized.In the fourth chapter,we obtain some new results:we proved that with sparse homogeneous parts,the polynomial automorphism satisfying ?2=I can be conjugated to its linear part;the polynomial retracts with sparse homogeneous parts is expanded,in the condition of c ?0,the polynomial map ? that satisfies ?2=c? is characterized,and we proved that the image of ? is isomorphic to a polynomial algebra.Theorem 0.1.Let k be a field of characteristic 0,and ? is an endomorphism of k[n]with sparse homogeneous parts and ?2=I.So there exist ai,bi ? Z+,i=1,…,t,such as where 2?ai?bi<ai+a1+1,i=1,…,t,(?),i=1,2,…,t-1.Then ? is conjugated to a linear endomorphism.Theorem 0.2.Let ? be an endomorphism of k[n]with sparse homogeneous parts,and ?2=c?,wherec ? k,c ?0.Then exists a tame automorphism such that?-1(?)?(?)?=(cx1+h'1,…,cxr+h'r,0,…,0).whereh'i?<xr+1,…,xn>.In particular,?(k[n])?k[r].