On some questions around Berest's conjecture

Abstract

Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_1^2$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_1^2$ is finite. Then the Dixmier conjecture holds, i.e $\forall \phi \in End(A_1)-\{0\}$, $\phi$ is an automorphism. Assume $\phi$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $\phi (P)=P$.

V3: 15 p, a significally elaborated version; the proof of the main result from V1 is greatly simplified, the second result is based on a part of the previous proof; important references added

References in corpus (5)

• Commuting ordinary differential operators with polynomial coefficients and automorphisms of the first Weyl algebra
• Commuting differential operators of rank 2 with polynomial coefficients
• Fourier-Mukai transform on Weierstrass cubics and commuting differential operators
• Commuting Ordinary Differential Operators and the Dixmier Test
• On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two