Abhyankar–Moh theorem

An affine algebraic variety (with an algebraically closed field of characteristic zero) is said to have the Abhyankar–Moh property if every imbedding extends to an automorphism of . The original Abhyankar–Moh theorem states that an imbedded affine line in has the Abhyankar–Moh property, [a1].

The algebraic version of this theorem (which works over any field) is as follows. Let be a field of characteristic . Let be such that . Let and . If , suppose in addition that does not divide . Then divides or divides .

If has small in comparison with and has "nice" singularities, then has the Abhyankar–Moh property [a2], [a4], [a5]. For every , the -cross has the Abhyankar–Moh property, [a3]. The case of a hyperplane in 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 to automorphisms of " 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