On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field

Mattarei, Sandro (2007) On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field. Finite Fields and Their Applications, 13 (4). pp. 773-777. ISSN 1071-5797

Full content URL: http://dx.doi.org/10.1016/j.ffa.2006.03.005

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Item Type:Article
Item Status:Live Archive

Abstract

In 1988 Garcia and Voloch proved the upper bound 4n^{4/3} (p−1){2/3} for the number of solutions over a prime finite field Fp of the Fermat equation x^n+y^n=a, where a∈F_p* and n⩾2 is a divisor of p−1 such that (n−1/2)^4⩾p−1. This is better than Weil's bound p+1+(n−1)(n−2) sqrt{p} in the stated range. By refining Garcia and Voloch's proof we show that the constant 4 in their bound can be replaced by 3⋅2^{−2/3}.

Keywords:Fermat equation, finite field
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:18506
Deposited On:12 Dec 2015 21:50

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