Parametrizing Algebraic Curves

TL;DR

This paper discusses the parametrization of plane algebraic curves, highlighting a classical proof that nonsingular curves of degree greater than two cannot be parametrized by rational functions, from a number theorist's perspective.

Contribution

It provides a clear presentation of Kapferer's proof and insights into the limitations of rational parametrization for certain algebraic curves.

Findings

Nonsingular curves of degree > 2 cannot be parametrized by rational functions.

Kapferer's proof is simple, elegant, and less known.

The paper offers a number theorist's perspective on curve parametrization.

Abstract

We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.