On the squarefree values of $a^4+b^3$

TL;DR

This paper proves that the density of integer pairs for which a specific polynomial is squarefree matches the predicted local densities, extending to a family of similar polynomials, with precise counting and error bounds.

Contribution

It establishes the asymptotic density of squarefree values of $a^4+b^3$ and related polynomials, confirming a conjectured product of local densities with exact counts and error estimates.

Findings

Density matches the conjectured product of local densities.

Exact count of pairs with squarefree polynomial values with power-saving error.

Results extend to polynomials $eta a^4 + heta b^3$ for fixed integers $eta, heta$.

Abstract

In this article, we prove that the density of integers $a, b$ such that $a^4+b^3$ is squarefree, when ordered by $\max\{|a|^{1/3},|b|^{1/4}\}$, equals the conjectured product of the local densities. We show that the same is true for polynomials of the form $\beta a^4 + \alpha b^3$ for any fixed integers $\alpha$ and $\beta$. We give an exact count for the number of pairs $(a,b)$ of integers with $\max\{|a|^{1/3},|b|^{1/4}\}<X$ such that $\beta a^4 + \alpha b^3$ is squarefree, with a power-saving error term.