Squareful numbers in hyperplanes

Van Valckenborgh, Karl

doi:10.2140/ant.2012.6.1019

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Squareful numbers in hyperplanes

Karl Van Valckenborgh

Vol. 6 (2012), No. 5, 1019–1041

DOI: 10.2140/ant.2012.6.1019

Abstract

Let $n \geq 4$ . In this article, we will determine the asymptotic behavior of the size of the set of integral points $(a_{0} : \dots : a_{n})$ on the hyperplane $\sum_{i = 0}^{n} X_{i} = 0$ in $ℙ^{n}$ 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 $| a_{i} | \leq B$ for each $i \in {0, \dots, n}$ , 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.

Keywords

squareful, Campana, asymptotic behavior

Mathematical Subject Classification 2010

Primary: 11D45

Secondary: 14G05, 11D72, 11P55

Milestones

Received: 3 December 2010

Revised: 17 June 2011

Accepted: 19 July 2011

Published: 31 July 2012

Authors

Karl Van Valckenborgh
Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200B 3001 Leuven Belgium

Vol. 6, No. 5, 2012