The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let An be the nth Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {Ancircle times operatorPm} is proved: an algebra A admits a finite set δ1,…,δs of commuting locally nilpotent derivations with generic kernels and image iff Asimilar, equalsAncircle times operatorPm for some n and m with 2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra Ancircle times operatorPm (and for image) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287–330]) given image, then image (the proof is algebro-geometric). We extend this result (using [non-holonomic] image-modules): given image, then image. Any automorphism image is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102–119], a similar result is proved for image.