Injectivity on one line

Abstract

Let $k$ be an algebraically closed field of characteristic zero. Let $H:k^2\to k^2$ be a polynomial mapping such that the Jacobian $\text{Jac}H$ is a non-zero constant. In this note we prove, that if there is a line $l\subset k^2$ such that $H{|}_l:l\to k^2$ is an injection, then $H$ is a polynomial automorphism.

2 pages, AmS-TeX

Cited by in corpus (7)

• Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
• Some remarks on the Jacobian conjecture and polynomial endomorphisms
• Plane automorphisms given by polynomials of scattered degrees
• Integer points on a curve and the plane Jacobian problem
• Siegel's theorem on integral points and the Jacobian conjecture over the rational field
• A variation on Magnus' theorem and its generalizations
• Two remarks on non-zero constant Jacobian polynomial map of $C^2$