Algebra & Number Theory

Squareful numbers in hyperplanes

Karl Van Valckenborgh

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Let n4. 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.

Article information

Algebra Number Theory, Volume 6, Number 5 (2012), 1019-1041.

Received: 3 December 2010
Revised: 17 June 2011
Accepted: 19 July 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D45: Counting solutions of Diophantine equations
Secondary: 14G05: Rational points 11D72: Equations in many variables [See also 11P55] 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

squareful Campana asymptotic behavior


Van Valckenborgh, Karl. Squareful numbers in hyperplanes. Algebra Number Theory 6 (2012), no. 5, 1019--1041. doi:10.2140/ant.2012.6.1019.

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