Let $f(x) = \prod_{i=0}^{n-1}(x - \alpha_i)$ be a polynomial over finite field. Assume there exists $0 \leq j < n$ s.t. $\operatorname{ord}(\alpha_j) > n$. Given only roots $\alpha_i$ being distinct, what is the formula for the number of distinct nonzero coefficients of $f(x)$?
Some special case I am interesting in. Let $\alpha_i = \beta^{t_i}$ for some $t_i$ and $\beta$ s.t. $\operatorname{ord}(\beta) > n$. So $f(x)$ is the generating polynomial of the Reed-Solomon code of length $m > n$ and $\beta$ is a primitive $m$-th root of unity. E.g. $f(x) = (x-1)(x-\beta)(x-\beta^6) = x^3 + (1+\beta^2) x^2 + (1+\beta^2) x + 1$ for $\beta$ being a primitive element of $\mathcal{F}_8 \equiv \mathcal{F}_2[x]/\langle x^3 + x^2 + 1 \rangle$ has only $2$ distinct coefficients.