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

TL;DR

This paper investigates the distribution of roots of random polynomials over p-adic fields, providing asymptotic formulas for moments and bounds on the probability of many roots.

Contribution

It introduces new asymptotic formulas for factorial moments and bounds on the probability of having many roots, advancing understanding of roots over p-adic numbers.

Findings

Asymptotic formula for factorial moments of roots

Probability bound for more than log n roots

Roots distribution behavior over p-adic fields

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 $K > 0$.