The existence of Fq-primitive points on curves using freeness

Abstract

Let CQ be the cyclic group of orderQ, n a divisor ofQ and r a divisor ofQ/n.We introduce the set of (r,n)-free elements of CQ and derive a lower bound for the number of elements 2 Fq for which f ( ) is (r,n)- free and F( ) is (R,N)-free, where f ,F 2 Fq [x]. As an application, we consider the existence of Fq -primitive points on curves like yn f (x) and find, in particular, all the odd prime powers q for which the elliptic curves y2 x3 x contain an Fq -primitive point. © 2022 Elsevier Masson SAS. All rights reserved.