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

Let AnAn be the nnth Weyl algebra   and PmPm be a polynomial algebra in mm variables over a field KK of characteristic zero. The following characterization of the algebras {An⊗Pm}{An⊗Pm} is proved: an algebra  AAadmits a finite set  δ1,…,δsδ1,…,δsof commuting locally nilpotent derivations with generic kernels and  ∩i=1sker(δi)=Kiff  A≃An⊗PmA≃An⊗Pmfor some  nnand  mmwith  2n+m=s2n+m=s, and vice versa. The inversion formula   for automorphisms of the algebra An⊗PmAn⊗Pm (and for P̂m≔K〚x1,…,xm〛) 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  σ∈AutK(Pm), then  degσ−1≤(degσ)m−1 (the proof is algebro-geometric  ). We extend this result (using [non-holonomic] DD-modules): given  σ∈AutK(An⊗Pm), then  degσ−1≤(degσ)2n+m−1. Any automorphism σ∈AutK(Pm) 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 σ∈AutK(An⊗Pm).One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209–242. [6]] problem 1) into a single question, (JD): is a  KK-algebra endomorphism  σ:An⊗Pm→An⊗Pmσ:An⊗Pm→An⊗Pman algebra automorphism provided  σ(Pm)⊆Pmσ(Pm)⊆Pmand  det(∂σ(xi)∂xj)∈K∗≔K∖{0}? (Pm=K[x1,…,xm])(Pm=K[x1,…,xm]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer   (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and pp-curvatures, Osaka J. Math. 42(2) (2005) 435–452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].