On the Abhyankar-Moh inequality

Abstract

In their fundamental paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In this note we study the semigroups of integers satisfying the Abhyankar–Moh inequality and describe such semigroups with the maximum conductor. Introduction. In this note we study the semigroups of integers appearing in connection with the Abhyankar–Moh inequality which is the main tool in proving the famous embedding line theorem (see [1, Main theorem]). Since the Abhyankar–Moh inequality can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve, it is natural to consider the semigroups for which such an inequality holds. Section 1 being devoted to preparatory lemmas, in Section 2 we study such semigroups (we call them Abhyankar–Moh semigroups). Then, in Section 3 we give a simplified proof of the Abhyankar–Moh Embedding Line Theorem. In what follows we need some basic properties of semigroups of the naturals. A subset G of N is a semigroup if it contains 0 and it is closed under addition. Let G be a nonzero semigroup and let n ∈ G, n > 0. Then there exists a unique sequence (v0, . . . , vh) such that v0 = n, vk = min(G\v0N + · · · + vk−1N) for 1 ≤ k ≤ h and G = v0N + · · · + vhN. We call the sequence (v0, . . . , vh) the n-minimal system of generators of G. If n = min(G\{0}) then we say that 2010 Mathematics Subject Classification. Primary 14R10; Secondary 32S05.