The existence of 𝔽 q -primitive points on curves using freeness

Abstract

Let 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 𝒞 Q and derive a lower bound for the number of elements θ∈𝔽 q for which f(θ) is (r,n)-free and F(θ) is (R,N)-free, where f,F∈𝔽 q [x]. As an application, we consider the existence of 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 ±x contain an 𝔽 q -primitive point.