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

Let An be the n’th Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {An⊗Pm} is proved: an algebra A admits a finite set δ1, . . . , δs of commuting locally nilpotent derivations with generic kernels and ∩i=1ker(δi) = K iff A ≃ An ⊗ Pm for some n and m with 2n + m = s, and vice versa. The inversion formula for automorphisms of the algebra An ⊗ Pm (and for P̂m := K[[x1, . . . , xm]]) has found (giving a new inversion formula even for polynomials). Recall that (see [3]) given σ ∈ AutK(Pm), then deg σ −1 ≤ (deg σ)m−1 (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given σ ∈ AutK(An ⊗ Pm), then deg σ−1 ≤ (deg σ)2n+m−1. Any automorphism σ ∈ AutK(Pm) is determined by its face polynomials [8], 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 [6] problem 1) into a single question, (JD): is a Kalgebra endomorphism σ : An ⊗ Pm → An ⊗ Pm an algebra automorphism provided det(i ∂xj ) ∈ K ∗ := K\{0}? (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 it follows from the recent paper [5])]. Mathematics subject classification 2000: 13N10, 13N15, 14R15, 14H37, 16S32.