Abhyankar–Moh theorem

An affine algebraic variety $X \subset k^n$ (with $k$ an algebraically closed field of characteristic zero) is said to have the Abhyankar–Moh property if every imbedding $\phi : X \rightarrow k^n$ extends to an automorphism of $k^n$. The original Abhyankar–Moh theorem states that an imbedded affine line in $k^2$ has the Abhyankar–Moh property, [a1].

The algebraic version of this theorem (which works over any field) is as follows. Let $k$ be a field of characteristic $p \ge 0$. Let $f,g \in k[T] \setminus k$ be such that $k[f,g] = k[T]$. Let $n = \deg f$ and $m = \deg g$. If $p > 0$, suppose in addition that $p$ does not divide $\mathrm{hcf}(f,g)$. Then $m$ divides $n$ or $n$ divides $m$.

If $X \subset \mathbb{C}^n$ has $\dim X$ small in comparison with $n$ and has "nice" singularities, then $X$ has the Abhyankar–Moh property [a2], [a4], [a5]. For every $n$, the $n$-cross $\{x \in \mathbb{C}^n : x_1\cdots x_n = 0 \}$ has the Abhyankar–Moh property, [a3]. The case of a hyperplane in $\mathbb{C}^n$ is still open (1998).

References

[a1]	S.S. Abhyankar, T-t. Moh, "Embeddings of the line in the plane" J. Reine Angew. Math. , 276 (1975) pp. 148–166
[a2]	Z. Jelonek, "A note about the extension of polynomial embeddings" Bull. Polon. Acad. Sci. Math. , 43 (1995) pp. 239–244
[a3]	Z. Jelonek, "A hypersurface that has the Abhyankar–Moh property" Math. Ann. , 308 (1997) pp. 73–84
[a4]	S. Kalliman, "Extensions of isomrphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$" Proc. Amer. Math. Soc. , 113 (1991) pp. 325–334
[a5]	V. Srinivas, "On the embedding dimension of the affine variety" Math. Ann. , 289 (1991) pp. 125–132