Given a polynomial $$ f(x) = a_nx^n + \ldots + a_1x + a_0 \in \mathbb C[x] $$ then if we fix $a_n, \ldots , a_1$, there are only finitely many values of $a_0$ such that $f(x)$ has repeated roots (corresponding to solutions of the discriminant of $f(x)$).
This is not true in $\bar {\mathbb F}_p$ - for example, one can take the polynomial $x^p+a$, which has all $p$ roots being the same for any $a\in \bar{\mathbb F}_p$.
My question is whether there is any way of characterising when the coefficients $a_n, \ldots , a_1$ (to use the same notation as in the complex example above) mean that $f(x)$ will have repeated roots for only finitely many values of $a_0$.
My guess would be that the only non-zero coefficients can be powers of $p$ (i.e. $a_n =0$ unless $n=p^k$ for some $k \in \mathbb N$). But this is just a guess, and I wouldn't want to comment on whether it is sufficient.
Hints:
Over any field $F$, a polynomial $p(x)\in\Bbb F[x]$ has a multiple root $\alpha$ iff $p(\alpha)=p'(\alpha)=0$
If $\Bbb F$ has characteristic $p>0$, the above is true iff the only non-zero coefficients of $p(x)$ are the ones of the form
$$a_{kp}, k\in\Bbb N \iff p(x)=\sum_{k=0}^n a_{kp}x^{kp}=g(x^p),g(x)\in\Bbb F[x]$$