×

Squareful numbers in hyperplanes. (English) Zbl 1321.11038

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

MSC:

11D45 Counting solutions of Diophantine equations
14G05 Rational points
11D72 Diophantine equations in many variables
11P55 Applications of the Hardy-Littlewood method
PDFBibTeX XMLCite
Full Text: DOI arXiv Link