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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).

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.

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.

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 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 $F(t,t)=(t,t-t)=(t,0)$.

Thank you very much for any comments and hints.

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 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.

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 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 $l=(x,\lambda)$, $\lambda \in \mathbb{R}$. This leads me to ask the following question:

Is Abhyankar-Moh theorem 1.6 still valid if we remove the algebraic closedness assumption? If I am not wrong AM theorem 1.6 appears as Theorem 1 of van den Essen's paper. I do not see where algebraic closedness is needed in the proof.

Thank you very much for any comments and hints.

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 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 $F(t,t)=(t,t-t)=(t,0)$.

Thank you very much for any comments and hints.