Let $p$ be a prime and $k,n$ integers $\geq 1$. Let $f(X)$ be a univariate degree $n$ polynomial in $(\mathbb{Z}/p^{kn}\mathbb{Z})[X]$ such that not all coefficients are divisible by $p$.
Is it possible for $f(X)$ to have more than $p^{k(n-1)}$ distinct zeros in $(\mathbb{Z}/p^{kn}\mathbb{Z})$? If so, is there an upper bound on the number of zeros such a polynomial can have?
I don't think the upper bound can be any smaller because of the following example. The polynomial $X^n$ has $p^{k(n-1)}$ zeros in $(\mathbb{Z}/p^{kn}\mathbb{Z})$ (the set $p^k(\mathbb{Z}/p^{kn}\mathbb{Z})$).