On some questions around Berest's conjecture

2022/03/24 by Junhu Guo, Alexander Zheglov, Guo, Junhu +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2203.13343

Abstract

Let K be a field of characteristic zero, let A₁=K[x][∂ ] be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial f(X,Y)∈ K[X,Y], which has a non-trivial solution (P,Q)∈ A₁² with [P,Q]=0, and the number of orbits under the group action of Aut(A₁) on solutions of f in A₁² is finite. Then the Dixmier conjecture holds, i.e ∀ φ∈ End(A₁)-\0\, φ is an automorphism. Assume φ 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∈ A₁, P∉ K, s.t. φ(P)=P.

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