m at h . A G ] 1 A ug 2 00 6 On the equivalence of the Jacobian , Dixmier and Poisson Conjectures

Jacobian Conjecture in characteristic p ≥ 0 [37] [41] [2] means that for any positive integer n, any jacobian 1 endomorphism of the algebra of polynomials in n indeterminates over a field of characteristic p is an automorphism, provided it induces a field extension of degree not a multiple of p. Dixmier Conjecture in characteristic p [26] means that, for any positive integer n, any endomorphism of the n-th Dirac quantum algebra over a field of characteristic p [23] [38] [20], unjustly called Weyl algebra by J. Dixmier in [26], i.e. the associative algebra over this field with 2n generators satisfying the normalized famous commutation relations of quantum mechanics, i.e. in other terms the algebra of “formal” differential operators in n indeterminates with polynomials coefficients over this field [1], is an automorphism, provided its restriction to the center of this Dirac algebra induces a field extension of degree not a multiple of p, and the jacobian of this restriction is a non zero element of the field in the case where p ≤ n. Poisson Conjecture in characteristic p means that for any positive integer n, any endomorphism of the n-th canonical Poisson algebra over a field of characteristic p , i.e. the algebra of polynomial in 2n indeterminates over this field endowed with its classical Poisson bracket, is an automorphism, provided it induces a field extension of degree not a multiple of p, and its jacobian of is a non zero element of the field in the case where p ≤ n. Thanks to recent results on ring homomorphisms of Azumaya algebras [6] and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these conjectures in characteristic zero on the model of [3], we prove the equivalence of these three conjectures in any characteristic, giving also by this way thanks to [6] a new proof of the equivalence of the complex version of the two first conjectures recently proved by Y. Tsuchimoto in a series of two papers [46] and