2015/09/21 by Vered Moskowicz, Moskowicz, Vered
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1509.06362
openalex publication_date 2015/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The famous Jacobian problem asks: Is a morphism f:ℂ[x,y]→ ℂ[x,y] having an invertible Jacobian, invertible? If we add the assumption that ℂ(f(x),f(y))=ℂ(x,y), then f is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If f:ℂ[x,y]→ ℂ[x,y] is a morphism having an invertible Jacobian, and if there exist n ≥ 1, a ∈ ℂ(f(x),f(y))^* and b ∈ ℂ(f(x),f(y)) such that (ax +b)n ∈ ℂ(f(x),f(y)), then f is invertible. A similar result holds for ℂ[x1,…,xm].