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 $O\left(n^{-K}\right)$ for some $K>0$.