Squarefree values of polynomials over the rational function field

TL;DR

This paper investigates the distribution of squarefree values of polynomials over finite fields, demonstrating that under certain conditions, most polynomial evaluations are squarefree as the field size grows.

Contribution

It extends the classical integer problem to function fields, proving that most polynomial evaluations are squarefree over large finite fields under specified conditions.

Findings

As the finite field size increases, most polynomial evaluations are squarefree.

The result applies to separable polynomials with square-free content, bounded degree, and height.

Almost all monic polynomials yield squarefree values when substituted into the polynomial.

Abstract

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then as the finite field size #F tends to infinity, for almost all monic polynomials a(t), the polynomial f(a) is squarefree.