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

Given a positive integer $Q$, denote by $mathcal{C}_Q$ the multiplicative cyclic group of order $Q$. Let $n$ be a divisor of $Q$ and $r$ a divisor of $Q/n$. Guided by the well-known formula of Vinogradov for the indicator function of the set of primitive elements of a finite field $mathbb{F}_q$, we derive an expression for the indicator function for the set of $(r,n)$-free elements of $mathcal{C}_Q$, i.e., the subset of the subgroup $mathcal{C}_{Q/n}$ comprising elements that are $r$-free in $mathcal{C}_{Q/n}$, i.e., are not $p$-th powers in $mathcal{C}_{Q/n}$ for any prime $p$ dividing $r$. We deduce a general 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 $f, F inmathbb{F}_q[x]$ and $n,N$ are divisors of $q-1$ with $rmid (q-1)/n$, $Rmid (q-1)/N$. As an application, we consider the existence of $mathbb{F}_q$-primitive points (i.e., points whose coordinates are primitive elements) on curves like $y^n=f(x)$. In particular, elliptic curves $y^2=f(x)$, where $f$ is a square-free cubic, are studied. We find, for example, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 pm x$ contain an $mathbb{F}_q$-primitive point.