Segre's theorem on ovals in Desarguesian projective planes

TL;DR

This paper provides a complete, self-contained, and simplified proof of Segre's theorem on ovals in Desarguesian projective planes, accessible to undergraduates with basic algebra background.

Contribution

It offers a shorter, clearer proof of Segre's theorem, including detailed explanations of prerequisites like homogeneous polynomials and Desargues' theorem.

Findings

Proof is shorter and simpler than original

Accessible to undergraduates with basic algebra

Includes detailed prerequisites and explanations

Abstract

Segre's theorem on ovals in projective spaces is an ingenious result from the mid-twentieth century which requires surprisingly little background to prove. This note, suitable for undergraduates with experience of linear and abstract algebra, provides a complete and self-contained proof. All necessary pre-requisites, principally evaluation of homogeneous polynomials at projective points and Desargues' theorem are presented in full. While following the broad outline of Segre's proof, careful parameterisation of certain tangent lines results in shorter and simpler computations than the original.