Squareful numbers in hyperplanes

TL;DR

This paper investigates the asymptotic count of integral points with squareful coordinates on a hyperplane in projective space, using Hardy-Littlewood method, with implications for the Brauer-Manin conjecture on Fano orbifolds.

Contribution

It provides the first asymptotic formula for the number of squareful integral points on hyperplanes, extending the understanding of rational points on Fano varieties.

Findings

Derived the asymptotic behavior of squareful points count as B increases.

Supports potential generalization of Brauer-Manin conjecture to Fano orbifolds.

Employs Hardy-Littlewood method for counting integral solutions.

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.