Let $F$ be the finite field with $q$ elements, where $q$ is a power of a prime, and let $E$ be its degree $n \geq 2$ extension. Let $f(x) \in F[x]$ such that $f(E) = F$. Clearly the number of distinct roots of $f(x)$ which lie in $E$ is at most $\deg(f(x))$.
In the special case when many of the monomials of $f(x)$ are of the form $x^{qk}$, $k \in \mathbb{N}$, say including the leading monomial, can we give a better general upper bound for the number of distinct roots of $f(x)$ in $E$? Thanks.