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

  • $\begingroup$ I switched one tag, for in the interesting cases we exit the realm of fields. $\endgroup$ Jul 20, 2020 at 6:07
  • $\begingroup$ 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. $\endgroup$ Jul 20, 2020 at 6:10
  • $\begingroup$ Nope, I think the $n$ in the exponent of $p^{kn}$ is killjoy. $\endgroup$ Jul 20, 2020 at 6:18


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