The Dixmier conjecture and the shape of possible counterexamples. (Q397880)

The Dixmier conjecture states that if \(k\) is an algebraically closed field of characteristic zero then any endomorphism of the Weyl algebra \(A_1(k)\) is an automorphism. The generalized Dixmier conjecture states the same for any Weyl algebra \(A_n(k)\). L. Vaserstein and V. Kac showed that the generalized conjecture implies the Jacobian conjecture. Y. Tsuchimoto, A. Belov-Kanel and M. Kontsevich, P. K. Adjamagbo and A. van den Essen proved that the Dixmier and Jacobian conjectures are stably equivalent.NEWLINENEWLINE Suppose that there exists a counterexample to the Dixmier conjecture, namely elements \(P,Q\in A_1(k)\) such that \(PQ-QP=1\) and \(P,Q\) do not generate the algebra \(A_1(k)\). Let \(B=\infty\) if Dixmier conjecture is true. Otherwise let \(B\) be equal to minimal greatest common divisor of total degrees of \(P,Q\), where the pair \(P,Q\) is a counterexample. The main result of the paper shows that \(B>15\).NEWLINENEWLINE The proof is rather technical. It is based on detailed analysis of various parameters associated with degrees of elements of \(A_1(k)\).