Fermat curves over finite fields and cyclic subsets in high-dimensional projective spaces
type: Item
instance of: scholarly article
title: Fermat curves over finite fields and cyclic subsets in high-dimensional projective spaces
zbMATH Open document ID: 0981.11022
author: Gabor Korchmáros
author: Tamás Szőnyi
published in: Finite Fields and their Applications
publication date: 2002-03-08T00:00:00Z
review text: The authors prove the following theorem: Let \(q\) be a power of the prime \(p\), \(N=(q^m-1)/(q-1), n\) a divisor of \(N\) and \(d=\gcd(N/n,t+1),\) where \(t\equiv q\bmod 0 N/n\), \(0<t<N/n.\) Assuming that \[ p> \left({2\over\root t+1 \of{\sin(n\pi/(2N))}}+1\right)^{(t-1)(N/n-d)}, \] the number of \(\text{GF}(q^m)\)-rational points of the Fermat curve \(X^n+Y^n+Z^n=0\) is \[ 3n+n^2(q-2)+(d-1)(d-2). \] This interesting theorem is proved using methods and results from finite geometry and combinatorics. More specific: The authors study a set \(\Omega_n\) inside the projective space \(\text{PG}(m-1,q)\) (\(\Omega_n\) is a Singer cycle of size \(N/n\)). The proof is based on counting the number of intersection points a line can have with \(\Omega_n\). From this the above formula is derived by looking at the case where only the numbers \(0,1,2,d\) occur. The authors show that the general approach can also be used to find explicit formulas in cases not covered by the theorem.
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 1295221
zbMATH Keywords: Fermat curve
zbMATH Keywords: rational points
zbMATH Keywords: cyclic sets in projective space
zbMATH Keywords: caps
Wikidata QID: Q59249264
reviewed by: Hubert Kiechle
MaRDI profile type: Publication
full work available at URL: ffta.1998.0242
OpenAlex ID: W2069189880
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DOI: 10.1006/FFTA.1998.0242
description: scientific article; zbMATH DE number 1295221