The uniqueness of a plane curve of degree q attaining Sziklaiʼs bound over Fq

We prove the uniqueness of a plane curve of degree q over a finite field Fq which attains Sziklaiʼs bound q(q−1)+1. More precisely, if a plane curve of degree q over Fq has q(q−1)+1 rational points, then it is projectively equivalent to the curve defined by the equation Xq−XZq−1+Xq−1Y−Yq=0. Although the case q=4 is the exception to Sziklaiʼs bound, the uniqueness of a curve of degree 4 with 13 points over F4 still holds.