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

Abstract: In this article, we prove that the density of integers $a, b$ such that $a<sup>4+b<sup>3$ is squarefree, when ordered by $\max{|a|<sup>{1/3},|b|<sup>{1/4}}$, equals the conjectured product of the local densities. We show that the same is true for polynomials of the form $\beta a⁴ + \alpha b<sup>3$ for any fixed integers $\alpha$ and $\beta$. We give an exact count for the number of pairs $(a,b)$ of integers with $\max{|a|<sup>{1/3},|b|<sup>{1/4}}&lt;X$ such that $\beta a⁴ + \alpha b<sup>3$ is squarefree, with a power-saving error term.