After asking [this question](https://mathoverflow.net/questions/295158/generalizations-of-abhyankar-moh-theorem-embeddings-of-the-line-in-the-plane/295575?noredirect=1#comment739839_295575), I figured out that I am also interested in the following related question:



> 
> Is [Abhyankar-Moh theorem 1.6](https://eudml.org/doc/151610) still valid if we remove the algebraic closedness assumption? 
> 
> 
> 


The answer is probably no, but I do not know how to find a counterexample over $\mathbb{R}$, for example.


Moreover, if I am not wrong, AM theorem 1.6 appears as Theorem 1 of [van den Essen's paper](https://link.springer.com/chapter/10.1007/978-3-642-18487-1_17); I do not see where algebraic closedness is needed in the proof.


Thank you very much for any comments or hints!


**Edit:** I have recently found [this](https://arxiv.org/pdf/1012.3064.pdf) beautiful criterion (Theorem 3.3), which does not require algebraic closedness of the base field. 



Abhyankar-Moh embedding theorem without algebraic closedness
In the van den Essen's paper, a polynomial map $t\mapsto (x(t),y(t))$ is called an embedding if there is a polynomial $F$ such that $F(x(t),y(t))=t$. There is no problem with his proof: Theorem 1 is true over any field if we interpret embedding in this sense. 


The reason why he considered $\mathbb{C}$ rather then an arbitrary field is that the above *definition* is weird if used for not algebraically closed fields such as reals. (In this case, one might call it something like "strong embedding" or "nice embedding", but just "embedding" would be quite misleading.) 


