The uniqueness of a plane curve of degree q attaining Sziklai's bound over F-q

Abstract

We prove the uniqueness of a plane curve of degree q over a finite field F-q which attains Sziklai's bound q(q - 1) + 1. More precisely, if a plane curve of degree q over F-q has q(q 1) + 1 rational points, then it is projectively equivalent to the curve defined by the equation X-q - XZ(q-1) + Xq-1Y - Y-q = 0. Although the case q = 4 is the exception to Sziklai's bound, the uniqueness of a curve of degree 4 with 13 points over F-4 still holds. (C) 2011 Elsevier Inc. All rights reserved.