A Strong Abhyankar-Moh Theorem and Criterion of Embedded Line

TL;DR

This paper strengthens the Abhyankar-Moh Theorem by weakening its conditions and provides a new criterion for identifying embedded lines based on polynomial derivatives.

Contribution

It introduces a stronger version of the Abhyankar-Moh Theorem and derives a novel criterion for embedded lines using derivatives of polynomials.

Findings

Weaker conditions suffice for a polynomial curve to be a line.

A criterion for embedded lines based on derivatives is established.

The theorem's applicability is extended to broader polynomial cases.

Abstract

The condition of plane polynomial curve to be a line in well-known Abhyankar-Moh Theorem is replaced by weaker ones. A criterion of embedded line is obtained from this strong theorem: Two polynomials can generate the entire polynomial ring iff their derivatives can be generated.