2016/05/31 by Vered Moskowicz, Moskowicz, Vered · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Commutative Algebra (math.AC) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1606.00426
openalex publication_date 2016/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: ℂ[x,y] → ℂ[x,y] be a ℂ-algebra endomorphism having an invertible Jacobian. We show that for such f, if, in addition, the group of invertible elements of ℂ[f(x),f(y),x][1/v] ⊂ ℂ(x,y) is contained in ℂ(f(x),f(y))-0, then f is an automorphism. Here v ∈ ℂ[f(x),f(y)]-0 is such that y = u/v, with u ∈ ℂ[f(x),f(y),x]-0. Keller's theorem (in dimension two) follows immediately, since Keller's condition ℂ(f(x),f(y))=ℂ(x,y) implies that the group of invertible elements of ℂ[f(x),f(y),x][1/v] is contained in ℂ(x,y)-0 = ℂ(f(x),f(y))-0.