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

Let A(n) be the nth Weyl algebra and P-m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A(n) circle times P-m} is proved: an algebra A admits a finite set delta(1),..., delta(s) of commuting locally nilpotent derivations with generic kernels and boolean AND(s)(i=1) ker(delta(i)) = K iff A similar or equal to A(n) circle times P-m for some n and m with 2n + m = s, and vice versa. The inversion formula for automorphisms of the algebra A(n) circle times P-m (and for (P) over cap (m) := K[[x(l),..., x(m)]]) 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 sigma is an element of Aut(K) (P-m), then deg sigma(-1) <= (deg sigma)m(-1) (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given sigma is an element of Aut(K)(A(n) circle times P-m), then deg sigma(-1) <= (deg sigma)(2n+m-1). Any automorphism sigma is an element of Aut(K)(P-m) 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 sigma is an element of Aut(K) (A(n) circle times P-m). One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algebres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is a K-algebra endomorphism sigma : A(n) circle times P-m -> A(n) circle times P-m an algebra automorphism provided sigma (Pm) subset of Pm and det(partial derivative xj/partial derivative sigma(xi) is an element of K* := K \ {0}? (P-m = K[x(l),..., x(m)]). 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 p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [101] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv: math. RA/0512171. [5]])]. (c) 2006 Elsevier B.V. All rights reserved.