Abhyankar-Moh theorem says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane. It seems that one can replace $\mathbb{C}$ by any algebraically closed field $k$ of characteristic zero.

Are there 'similar' results for fields other than $k$, especially fields of characteristic zero, not necessarily algebraically closed?

Also, it was mentioned in Wikipedia that there are generalizations of this theorem to higher dimensions.

Can one please explain what those generalizations are or give a reference?

Any comments are welcome!


The general question that one can ask is if embeddings $\iota:\mathbb{A}^k\hookrightarrow\mathbb{A}^n$ can be rectified, i.e., if there is an automorphism of $\mathbb{A}^n$ which takes $\iota$ to the standard embedding of a coordinate $k$-plane. The Abhyankar-Moh(-Suzuki) theorem is the case where $k=1$ and $n=2$. The case where $k=n-1$ is the Abhyankar-Sathaye conjecture.

There are several things that could be called higher-dimensional generalization. For example, there are results of Craighero, Jelonek, Nori, Srinivas, etc... which show that embeddings between affine spaces are rectifiable if the codimension is high enough. Here are some references:

  • P. Craighero. A result on $m$-flats in $\mathbb{A}^n_k$. Rend. Sem. Math. Univ. Padova 75 (1986), 39-46.

  • Z. Jelonek. The extension of regular and rational embeddings. Math. Ann. 277 (1987), 113-120. (link to paper on Springer website)

  • V. Srinivas. On the embedding dimension of an affine variety. Math. Ann. 289 (1991), 125-132. (link to paper on Springer website)

The most general result, proved as Theorem 2 in the paper of Srinivas above (and attributed to Nori) states that an embedding $\mathbb{A}^k$ in $\mathbb{A}^n$ is rectifiable if $n\geq 2k+2$. Note that the base field for Srinivas' theorem can be any infinite field.

Embeddings of other subvarieties could also be considered:

  • S. Kaliman. Extensions of isomorphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$. Proc. Amer. Math. Soc. 113 (1991), 325–334. (link to paper on AMS website)

There is also survey of van den Essen with some more literature references on generalizations of Abhyankar-Moh and related questions:

  • A. van den Essen. Around the Abhyankar-Moh theorem. In: Algebra, arithmetic and geometry with applications (Papers from Shreeram S. Abhyankar's 70th birthday conference), Springer (2004), pp. 283-294. (link to paper on Springer website)
  • $\begingroup$ Please, what is $\mathbb{A}$? Is it ok to take $\mathbb{A}=\mathbb{R}$? $\endgroup$ – user237522 Mar 25 '18 at 5:02
  • $\begingroup$ $\mathbb{A}^n$ denotes the affine space of dimension $n$ (base field here is implicit). $\endgroup$ – Matthias Wendt Mar 26 '18 at 9:08
  • $\begingroup$ Thank you for answering me. So the base field can be an arbitrary field? (1) not necessarily algebraically closed? (2) not necessarily of characteristic zero? (3) an integral domain? (4) an arbitrary commutative ring? Is it true that the answers are: (1) yes. (2) yes. (3) no? or maybe considering its field of fractions may yield something interesting?. (4) no? $\endgroup$ – user237522 Mar 26 '18 at 10:23
  • $\begingroup$ @user237522: for Srinivas' result, the base field can be an arbitrary field, not necessarily closed, not necessarily characteristic zero. I am not aware of generalizations to rings. It would seem to me that this is a much more complicated question. $\endgroup$ – Matthias Wendt Mar 26 '18 at 21:34
  • $\begingroup$ Thank you very much! (I guess that you have meant in your last comment "for Srinivas' result the base field can be an arbitrary infinite field etc.", as you have mentioned in your answer). $\endgroup$ – user237522 Mar 26 '18 at 22:03

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