A remark on the two-dimensional Jacobian conjecture (Q1336816)

Let \(k\) be an algebraically closed field of characteristic zero. Let \(F = (f,g) : k^ 2 \to k^ 2\) be a polynomial map. It is shown that if \(\text{det} JF \in k^*\) and there exist three concurrent lines such that the restriction of \(F\) to each of these lines is injective, then \(F\) is invertible. A corollary is that if \(F\) is of the form \(F=X+H\) with \(H\) homogeneous of degree \(\geq 2\) and \(\text{det} JF \in k^*\), then \(F\) is invertible.