Vol. 6, No. 5, 2012

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Squareful numbers in hyperplanes

Karl Van Valckenborgh

Vol. 6 (2012), No. 5, 1019–1041
Abstract

Let n 4. In this article, we will determine the asymptotic behavior of the size of the set of integral points (a0 : : an) on the hyperplane i=0nXi = 0 in n such that ai is squareful (an integer a is called squareful if the exponent of each prime divisor of a is at least two) and |ai| B for each i {0,,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