Injectivity on one line

TL;DR

This paper proves that a polynomial map with constant non-zero Jacobian that is injective on some line in the plane must be a polynomial automorphism.

Contribution

It establishes a new criterion for polynomial automorphisms based on injectivity on a line, extending understanding of the Jacobian conjecture in two dimensions.

Findings

If H|_l is injective on a line l, then H is an automorphism.

The result applies over algebraically closed fields of characteristic zero.

Provides a new condition for polynomial automorphisms in the plane.

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.