Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Another Proof of Segre's Theorem about Ovals

Peter Müller

TL;DR

This note addresses Segre's theorem, which states that any oval in the projective plane over a finite field $F$ of odd order is a conic. It introduces a direct, algebraic route: prove Theorem $T$, which says that any map $f:F o F$ whose graph has no three collinear points must be a quadratic polynomial, and then derive Segre's theorem from this parametrization. The main contribution is a self-contained proof of Theorem $T$ using slope-counting and a symmetry argument to produce an explicit quadratic form for $f$, from which the oval is shown to be the conic $YZ = aX^2 + bXZ + cZ^2$. This provides an alternative route to Segre's characterization of ovals and enriches the understanding of incidence geometry over finite fields.

Abstract

In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points and tangents, and then shows that it actually coincides with the oval. The different proof given here parametrizes an affine piece of the oval and shows directly that the parametrization is given by a polynomial of degree $2$.

Another Proof of Segre's Theorem about Ovals

TL;DR

This note addresses Segre's theorem, which states that any oval in the projective plane over a finite field of odd order is a conic. It introduces a direct, algebraic route: prove Theorem , which says that any map whose graph has no three collinear points must be a quadratic polynomial, and then derive Segre's theorem from this parametrization. The main contribution is a self-contained proof of Theorem using slope-counting and a symmetry argument to produce an explicit quadratic form for , from which the oval is shown to be the conic . This provides an alternative route to Segre's characterization of ovals and enriches the understanding of incidence geometry over finite fields.

Abstract

In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points and tangents, and then shows that it actually coincides with the oval. The different proof given here parametrizes an affine piece of the oval and shows directly that the parametrization is given by a polynomial of degree .

Paper Structure

This paper contains 3 sections, 1 theorem, 9 equations.

Table of Contents

  1. Introduction
  2. Theorem \ref{['T']} implies Segre's theorem
  3. Proof of Theorem \ref{['T']}

Key Result

Theorem 1

Let $F$ be a finite field of odd order, and $f:F\to F$ be a map such that no three points of the graph $\{(x,f(x))\,\mid\,x\in F\}$ are collinear in the affine plane over $F$. Then $f$ is given by a quadratic polynomial.

Theorems & Definitions (1)

  • Theorem 1