On some questions around Berest's conjecture

Abstract: Let

K

be a field of characteristic zero, let

A_{1} = K [x] [p a r t i a l]

be the first Weyl algebra. In this paper we prove the following result. Assume there exists a non-zero polynomial

f (X, Y) i n K [X, Y]

, which has a non-trivial solution

(P, Q) i n A_{1}^{2}

(which implies

[P, Q] = 0

), and the number of orbits under the group action of

A u t (A_{1})

on solutions of

f

in

A_{1}^{2}

is finite. Then the Dixmier conjecture holds, i.e

f o r a l l v a r p h i i n E n d (A_{1}) - 0

,

v a r p h i

is an automorphism.

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