The Number of Roots of a Random Polynomial over The Field of $p$-adic Numbers

Abstract

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this polynomial. In addition, we show the probability that a random polynomial of degree $n$ has more than $\log n$ roots is $O\big(n^{-K}\big)$ for some $K > 0$.