Bounding the number of polynomials in $\mathbb{F}_p$ where its discriminant is divisible by some fix prime number $p$.

Question

How can I find the upper bound for the number of monic polynomials $F(X) = X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{F}_p[X]$ such that its discriminant is congruent to $0$ mod $p$, where $p$ is a prime number and $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$? i.e.,

$\sum_{\substack{(a_1,\ldots,a_n) \in \mathbb{F}^n_p \\ \Delta(a_1,\ldots,a_n) \equiv 0 \ \text{mod} \ p}} 1$

where $\Delta(a_1,\ldots,a_n)$ defines the discriminant of $F(X) = X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{F}_p[X]$

Answer 1

Discriminant is zero iff the polynomial has a repeated factor, so you want the number of polynomials that are not square-free. Qiaochu Yuan derives the relevant generating function here.

Unless I misinterpreted that post tells us that there are exactly $p^{n-1}$ monic polynomials of degree $n$ in $\Bbb{F}_p[X]$ with discriminant zero.