Let $\Bbb F_{p^n}$ denote the field of $p^n$ elements and let $q(x)\in\Bbb F_{p^n}[x]$ be a polynomial of degree at most $t$. We know that $q(x)$ has at most $t$ roots, therefore, the ratio between the number of roots and the degree is at most $1$.

Now to the question

I am interested in analyzing the above result when we replace the field $\Bbb F_{p^n}$ by $\Bbb Z_{p^n}$, the ring of integers modulo $p^n$. Moreover, I would like to know the behavior of such ratio when $t$ depends on $n$ (say, polynomially) and $n$ goes to infinity.

More precisely, let $r_n$ be the maximum number of roots of all polynomials $q(x)\in\Bbb Z_{p^n}[x]$ of degree at most $n$ (here I choose $t(n) = n$ for concreteness, but this is not a requirement at all!). Since this ring has zero divisors, we can find polynomials of degree at most $n$ with many more roots than $n$. My goal is to study in more detail this "many more".

The question is:

What can we say about the ratio $\frac{r_n}{n}$ as $n$ goes to infinity? Does it go to infinity? Is it bounded?

Among other things, I considered Hensel's lemma. However, I don't see so much use on it as I'm not considering lifting of roots or anything similar here.

Any insights on this question will be very appreciated.

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    $\begingroup$ Hensel should apply. If you fix $q(x)\in\mathbb{Z}/p^n[x]$, then you could look at the reduction $\widetilde{q}(x)\in\mathbb{Z}/p[x]$, and lift roots of $\widetilde{q}(x)$ until you reach $\mathbb{Z}/p^n$. This would allow you to use Hensel. $\endgroup$ – Michael Burr Aug 26 '17 at 19:33
  • $\begingroup$ @MichaelBurr Thanks for your comment. Indeed, that's what I tried, but as far as I understand some of the conditions I would need to be able to lift a root is that $q'(r)\not\equiv 0\mod p$, so what can we say about the roots for which this condition is not satisfied? $\endgroup$ – Daniel Aug 27 '17 at 8:05
  • $\begingroup$ You might want to check out Hensel lifting on wikipedia. It has a section on what happens when $q'(r)\equiv 0\mod p$. You likely also need $r_n$ to depend on $p$. $\endgroup$ – Michael Burr Aug 27 '17 at 13:08
  • $\begingroup$ @MichaelBurr Thank you Michael. Indeed, they have a section on that, but notice the comment "if $f'(r)\equiv 0{\pmod {p}}$, then $0$, $1$, or several $s$ may exist", I'm looking for an upper bound that "several". I'll keep looking, thank you so much for your attention on this question. $\endgroup$ – Daniel Aug 28 '17 at 14:51

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