  The Jacobian conjecture in dimension $n$ asserts that any polynomial
endomorphism of $n$-dimensional affine space over a field of zero
characteristic, with the Jacobian equal 1, is invertible. The Dixmier
conjecture in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra
(the algebra of polynomial differential operators in $n$ variables) is
invertible. We prove that the Jacobian conjecture in dimension $2n$ implies the
Dixmier conjecture in rank $n$. Together with a well-known implication in the
opposite direction, it shows that the stable Jacobian and Dixmier conjectures
are equivalent. The main tool of the proof is the reduction to finite
characteristic.
  After the paper was finished we have learned that the main result was already
published by Y.Tsuchimoto in Osaka Journal of Mathematics Volume 42, Number 2
(June 2005). His proof is different.
