Square-free values of polynomials on average

TL;DR

This paper advances the understanding of the distribution of square-free values of polynomials by improving average-case results and dependence on polynomial height, using methods related to the Bateman-Horn and Chowla conjectures.

Contribution

It refines the average results for square-free values of polynomials, reducing the dependence between the polynomial height and the variable range, based on new adaptations of recent methods.

Findings

Improved bounds on the average number of square-free values for polynomials.

Reduced dependence between polynomial height and the range of x.

Extended the applicability of average-case results to higher polynomial heights.

Abstract

The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was able to show conditionally on the $abc$-conjecture that this conjecture is true for polynomials of arbitrarily large degrees. In 2013 Shparlinski proved that this conjecture holds on average over all polynomials of a fixed naive height, which was improved by Browning and Shparlinski in 2023. In this paper, we improve the dependence between $x$ and the height of the polynomial. We achieve this via adapting a method introduced in a 2022 paper by Browning, Sofos, and Ter\"av\"ainen on the Bateman-Horn conjecture, the polynomial Chowla conjecture, and the Hasse principle on average.