Approximate roots

TL;DR

The paper introduces the concept of approximate roots in algebraic structures, explores their properties, and applies this understanding to prove the Abhyankar-Moh embedding line theorem.

Contribution

It generalizes the notion of approximate roots to semiroots and demonstrates their application in algebraic geometry, particularly in singularity analysis and embedding theorems.

Findings

Most properties of approximate roots extend to semiroots.

Sequence of approximate roots reveals singularity structure.

Application to prove Abhyankar-Moh embedding line theorem.

Abstract

Given an integral domain $A$, a monic polynomial $P$ of degree $n$ with coefficients in $A$ and a divisor $p$ of $n$, invertible in $A$, there is a unique monic polynomial $Q$ such that the degree of $P-Q^{p}$ is minimal for varying $Q$. This $Q$, whose $p$-th power best approximates $P$, is called the $p$-th approximate root of $P$. If $f \in \mathbf{C}[[X]][Y]$ is irreducible, there is a sequence of characteristic approximate roots of $f$, whose orders are given by the singularity structure of $f$. This sequence gives important information about this singularity structure. We study its properties in this spirit and we show that most of them hold for the more general concept of semiroot. We show then how this local study adapts to give a proof of Abhyankar-Moh's embedding line theorem.