On Certain Probabilistic Properties of Polynomials over the Ring of $p$-adic Integers

TL;DR

This paper investigates probabilistic characteristics of polynomials over $p$-adic integers, including separability and coprimality, using Haar measure, and extends these results to polynomials modulo powers of $p$.

Contribution

It introduces the concept of strong coprimality for polynomials and computes related probabilities, generalizing previous results and providing a framework for further probabilistic analysis.

Findings

Probability that a monic polynomial is separable calculated.

Probability that two monic polynomials are strongly coprime determined.

Method allows extrapolation of probabilistic properties from modular polynomials.

Abstract

In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo powers of $p$.