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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).

Thank you very much for any comments and hints.

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  • $\begingroup$ Regarding Question 1, the Abhyankar–Moh theorem is valid over any field of characteristic zero, and there is even a version over positive characteristic fields that puts conditions on exponents; see the Main Theorem in Abhyankar and Moh's original paper. The result by J. Lang cited by Gwoździewicz does not need the field to be algebraically closed. Did you find where Gwoździewicz needs that assumption? $\endgroup$
    – Takumi Murayama
    Commented Mar 27, 2018 at 17:37
  • $\begingroup$ If I am not wrong, algebraic closedness is needed for the implication: $\gamma: t \mapsto (f(t),g(t))$ is injective and $(f'(t),g'(t)) \neq (0,0)$ for all $t \in k$ implies that $\gamma$ is an embedding. (Because, in that case, the D-resultant is a non-zero constant, and then $k[f(t),g(t)]=k[t]$). $\endgroup$
    – user237522
    Commented Apr 19, 2018 at 17:05

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