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]).

Next, consider the following claim:
Let $f: k[x_1,\ldots,x_n] \to k[x_1,\ldots,x_n]$, 
$(x_1,\ldots,x_n) \mapsto (f_1,\ldots,f_n)$,
be a $k$-algebra endomorphism having an invertible Jacobian
(= the determinant of the Jacobi matrix $\in k^\times$), $n \geq 3$.
If there is a hyperplane $L \subset k^n$ (= $L$ is of dimension $n-1$)
such that $f$ restricted to $L$ is an injection, then $f$ is an automorphism of $k[x_1,\ldots,x_n]$.

> **Question 2:** (I) Is the $n \geq 3$ case true? 
**My answer:** I guess that one will have to generalize Abhyankar-Moh theorem and find a similar property
for the Newton polytopes of $f_1,\ldots,f_n$, 
and perhaps then the $n \geq 3$ case can be proved.
Actually, the generalization of Abhyankar-Moh theorem I may want to rely on is Abhyankar-Sathaye conjecture; is there any progress on that conjecture?
> 
> (II) If (I) has a positive answer, can we replace $k$ by any field of characteristic zero? **My answer:** The answer should be the same for the $n=2$ case and for the $n \geq 3$ case.

**Remark:** [O. Hadas][4] dealt with Newton polytopes of automorphisms.


  [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://www.sciencedirect.com/science/article/pii/002240499190098M