On a number of rational points on a plane curve of low degree

Abstract

In Homma and Kim (2010), an upper bound of the number of rational points on a plane curve of degree d over F-q is found. Some examples attaining the bound are given in Homma and Kim (2010), whose degrees are q + 2, q+ 1, q, q - 1, root q-1 (when q is a square), and 2. In this paper, we consider an actual upper bound on such numbers for curves of low degree for q <= 7. Also we give explicit examples of curves attaining the sharp bound for each d and q. (C) 2017 Elsevier B.V. All rights reserved.