Square-free values of polynomials evaluated at primes over a function field

TL;DR

This paper investigates the distribution of primes in function fields for which a given square-free polynomial evaluated at these primes yields square-free values, establishing a limiting density result.

Contribution

It extends the understanding of square-free values of polynomials at primes to function fields, providing a limiting density result where previous integer results were limited.

Findings

Existence of a limiting density for primes with square-free polynomial values

Extension of classical integer results to function fields

Applicable to polynomials with arbitrary irreducible factors

Abstract

We study a function field version of a classical problem concerning square-free values of polynomials evaluated at primes. We show that for a square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density as $n\to \infty$ of primes $P \in \mathbb{F}_q[t]$ of degree $n$ such that $f(P)$ is square-free. Over the integers the analogous result is only known when all irreducible factors of $f$ have degree at most 3.