The Image Of Differential Operators On Polynomial Algebras

In 1939,Keller present the Jacobian conjecture.Many mathematicians have done a lot of research on this conjecture,but it has not been solved yet.In 2007,W.Zhao introduced the theory of Mathieu subspace and Image conjecture to study the Jacobian conjecture.In the first chapter of this thesis,we introduce the background and current situation of the study of Image conjecture.In the second and the third chapters of this thesis,we introduce the relationship between Mathieu subspace,Image conjecture,Vanishing conjecture,and the Jacobian Conjecture.And some known results about Image Conjecture are also introduced.In the final chapter of this thesis,we introduce some results obtained by ourselves about Image conjecture.In this part,we mainly prove that the Higher Order Image Conjecture holds for any dimension in the case of characteristic p,and we also show a special case of one-dimensional High Order Image Conjecture in the case of characteristic 0.The results are as follows.Theorem0.1Let k be a field,chark=p,A be a commutative k-algebra with 1,A[z]=A[z₁,z₂,...,z_n]be the polynomial algebra in n variables over A.a₁,a₂,...,a_n?A be a regular sequence of A.Let D be the set of differential operators:?₁-a₁,?₂-a₂,...,?_n-a_n,where?_i?A[?_z1,?_z2,...,?_zn],i=1,2,...,n,and the constant coefficient is 0.Then Im D:=??ⁿ_i=1??_i-a_i?A[z]is a Mathieu subspace of A[z].Theorem0.2 Let k be a field,char k=0,A be a commutative k-algebra with 1,A[z]the one-variable polynomial algebra over A.Let D=?_z²-a²,where a?A is a non-zero-divisor and Aa is a radical idea of A.Then Im D=??_z²-a²?A[z]is a Mathieu subspace of A[z].