On the Abhyankar-Moh inequality

TL;DR

This paper studies semigroups satisfying the Abhyankar-Moh inequality and provides a simple proof of the Abhyankar-Moh embedding theorem, advancing understanding of plane algebraic curve embeddings.

Contribution

It offers a new, simplified proof of the Abhyankar-Moh embedding theorem based on properties of semigroups satisfying the inequality.

Findings

Characterization of semigroups satisfying the Abhyankar-Moh inequality

A simplified proof of the Abhyankar-Moh embedding theorem

Enhanced understanding of plane algebraic curve embeddings

Abstract

Abhyankar and Moh in their fundamental paper on the embeddings of the line in the plane 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 give a simple proof of the Abhyankar-Moh embedding theorem.