The main result of J. Gwozdziewicz in this paper says the following: "Let $k$ be an algebraically closed field of characteristic zero, and let $f:k[x,y] \to k[x,y]$, $(x,y) \mapsto (p,q)$, be a $k$-algebra endomorphism having an invertible Jacobian, namely, $p_xq_y-p_yq_x \in k^\times$. If there is a line $l \subset k^2$ such that $f$ restricted to $l$ is an injection, then $f$ is an automorphism of $k[x,y]$".

The proof relies on a famous result of Abhyankar and Moh and on a property of Newton polygons of a Jacobian pair Theorem 2.1.

Question 1: Can we replace $k$ by any field of characteristic zero, not necessarily algebraically closed? for example $\mathbb{R}$. My answer: The answer may depend on whether Abhyankar-Moh theorem is valid over a field of characteristic zero, not necessarily algebraically closed, and this I do not know (please see my question).

Edit: If I am not wrong, the w.l.o.g. asumption in Gwozdziewicz theorem that the line is given by $y=0$ uses Abhyankar-Moh theorem 1.6, in which algebraic closedness is necessary. So if I wish to have Gwozdziewicz theorem over $\mathbb{R}$, then I need to be careful what is exactly the given line $l$ is; for example, we can take $l=(x,0)$ or $l=(x,\lambda)$, $\lambda \in \mathbb{R}$.