The existence of ${\mathbb{F}}_q$-primitive points on curves using freeness

Abstract

Let ${\mathcal{C}}_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of ${\mathcal{C}}_Q$ and derive a lower bound for the the number of elements $\theta \in {\mathbb{F}}_q$ for which $f(\theta )$ is $(r,n)$-free and $F(\theta )$ is $(R,N)$-free, where $f,F\in {\mathbb{F}}_q[x]$. As an application, we consider the existence of ${\mathbb{F}}_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3\pm x$ contain an ${\mathbb{F}}_q$-primitive point.