# Polynomial zeros modulo a prime power

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 $$p^k(\mathbb{Z}/p^{kn}\mathbb{Z})$$).

• I switched one tag, for in the interesting cases we exit the realm of fields. Jul 20, 2020 at 6:07
• May be the descending factorials are exceptional? The values of the monic polynomial $f_k(x):=\prod_{i=0}^{k-1}(x-i)$ are divisible by $k!$, so for example $x(x-1)(x-2)(x-3)$ vanishes everywhere modulo $8$. I'm not sure whether that leads to anything interesting here. It is known that the ideal of polynomials $\in\Bbb{Z}[x]$ vanishing everywhere modulo a given prime power is generated by the appropriate integer multiples of these. Jul 20, 2020 at 6:10
• Nope, I think the $n$ in the exponent of $p^{kn}$ is killjoy. Jul 20, 2020 at 6:18