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).

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. According to the answer to my above mentioned question, the generalization of Abhyankar-Moh theorem I may want to rely on is Abhyankar-Sathaye conjecture; is there any progress on that conjecture? Also according to the above answer, if instead of a hyperplane we will take $L \subset K^n$ of dimension $r$ such that $n \geq 2r+2$, then is it true that injectivity on such $L$ implies that $f$ is an automorphism? I guess no? (this is why I have originally taken a hyperplane).

(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 dealt with Newton polytopes of automorphisms.

Edit: This paper is relevant to my second question, especially Corollary 2.9 (notice that it deals with degree $\leq 3$).