Images Of Locally Nilpotent Derivations Of Polynomial Algebras

Affine algebraic geometry is a branch of algebraic geometry,mainly con-cerned with affine varieties similar to affine spaces.Derivations(especially lo-cally nilpotent derivations)are one of the important tools for studying affine algebraic geometry.This paper focuses on the image of derivations and the Jacobian conjecture in affine algebraic geometry.The first chapter is mainly about Jacobian conjecture,the image of differ-ential operator(derivation)and Mathieu-Zhao subspace.The second chapter introduces several equivalent forms of Jacobian conjecture,including Diximier conjecture,Poisson conjecture,vanishing conjecture and the relationship be-tween the images of differential operators(derivations)and Jacobian conjecture is introduced in detail.The third chapter introduces the research results on derivations in the existing literature,especially on the images of derivations of divergence zeros and locally nilpotent derivations.The fourth chapter is our own research results on derivations,depicting the images of some derivations of polynomial algebras in two variables.The specific results are as follows:Theorem 0.1 Let k be a field of characteristic zero,D be a locally nilpo-tent derivation of k[x,y]and I=(u(x,y))be a nonzero ideal of k[x,y],for some u(x,y)? k[x,y].Then DI is a Mathieu-Zhao subspace of k[x,y].Theorem 0.2 Let D=ya(?)x-xa(?)y,a ? N.Then ImD is a Mathieu-Zhao subspace of R[x,y].