Lifting of Polynomial Symplectomorphisms and Deformation Quantization

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize -- and reprove -- D. Anick's fundamental result on approximation of polynomial automorphisms, adapt it to the case of symplectomorphisms, and formulate the lifting problem. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.