On square-free values of large polynomials over the rational function field

TL;DR

This paper studies the distribution of square-free values taken by large polynomials over the rational function field, addressing specific cases like short interval densities and polynomial representations.

Contribution

It provides new results on the density and asymptotic counts of square-free polynomial values in the rational function field setting.

Findings

Density of square-free polynomials in short intervals determined

Asymptotic formulas for representing polynomials as sums involving square-free parts

Results extend understanding of polynomial value distributions over function fields

Abstract

We investigate the density of square-free values of polynomials with large coefficients over the rational function field $\mathbb{F}_q[t]$. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial $N$ as a sum of a small $k$-th power and a square-free polynomial.