The main result of J. Gwozdziewicz in [this paper][1] 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][2] and on a property of Newton polygons of a Jacobian pair [Theorem 2.1][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][3]).

**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 assumed. So if I wish to have Gwozdziewicz theorem over $\mathbb{R}$, then I need to be careful what exactly the given line $l$ is;
for example, we can take $l=(x,0)$ or $l=(x,\lambda)$, $\lambda \in \mathbb{R}$.
Another example is $l=(x,x)$, since taking $F:(x,y)\mapsto (x,y-x)$ yields
$F(t,t)=(t,t-t)=(t,0)$.
Actually, $F(t,at+b)=(t,0)$, where $F:(x,y) \mapsto (x,y-ax-b)$.

Thank you very much for any comments and hints.

  [1]: https://arxiv.org/abs/alg-geom/9305008
  [2]: https://eudml.org/doc/151610
  [3]: https://mathoverflow.net/questions/295158/generalizations-of-abhyankar-moh-theorem-embeddings-of-the-line-in-the-plane
  [4]: https://link.springer.com/chapter/10.1007/978-3-642-18487-1_17