A new proof of R\'edei's theorem on the number of directions

TL;DR

This paper presents a concise new proof of Rédéi's theorem on the number of directions determined by subsets in finite fields, utilizing a lemma by Kiss and the author, and extends to polynomial results over finite fields.

Contribution

The paper offers a novel, shorter proof of Rédéi's theorem and introduces an extension involving polynomials over finite fields.

Findings

Confirmed the minimal number of directions as either 1 or at least (p+3)/2.

Provided a simplified proof method using a key lemma.

Extended the result to polynomial contexts over finite fields.

Abstract

R\'edei and Megyesi proved that the number of directions determined by a $p$ element subset of $\mathbb{F}_p^2$ is either $1$ or at least $\frac{p+3}{2}$. The same result was independently obtained by Dress, Klin and Muzychuk. We give a new and short proof of this result using a Lemma proved by Kiss and the author. The new proof further on a result on polynomials over finite fields.