Squareful numbers in hyperplanes

Abstract

Let $n \geqslant 4$. In this article, we will determine the asymptotic behaviour of the size of the set $M(B)$ of integral points $(a_{0}:... :a_{n})$ on the hyperplane $\sum_{i=0}^{n}X_{i}=0$ in $\mathbf{P}^{n}$ 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), non-zero and $|a_{i}|\leq B$ for each $i \in \{0,...,n\}$, when $B$ goes to infinity. For this, I will use the classical Hardy-Littlewood method. The result obtained supports a possible generalization of the Brauer-Manin program to Fano orbifolds.