Squareful numbers in hyperplanes

Abstract

Let $n\ge 4$. In this article, we will determine the asymptotic behavior of the size of the set of integral points $\left(a_0:\cdots :a_n\right)$ on the hyperplane ${\sum }_{i=0}^n{X}_i=0$ in ${\mathbb{P}}^n$ such that $a_i$ is squareful (an integer $a$ is called squareful if the exponent of each prime divisor of $a$ is at least two) and $\left|a_i\right|\le B$ for each $i\in \left\{0,\dots ,n\right\}$, when $B$ goes to infinity. For this, we will use the classical Hardy–Littlewood method. The result obtained supports a possible generalization of the Batyrev–Manin program to Fano orbifolds.