  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.
