Algebra & Number Theory

Squareful numbers in hyperplanes

Karl Van Valckenborgh

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729844

Digital Object Identifier
doi:10.2140/ant.2012.6.1019

Mathematical Reviews number (MathSciNet)
MR2968632

Zentralblatt MATH identifier
1321.11038

Subjects
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]

Keywords
squareful Campana asymptotic behavior

Citation

Van Valckenborgh, Karl. Squareful numbers in hyperplanes. Algebra Number Theory 6 (2012), no. 5, 1019--1041. doi:10.2140/ant.2012.6.1019. https://projecteuclid.org/euclid.ant/1513729844


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References

  • D. Abramovich, “Birational geometry for number theorists”, pp. 335–373 in Arithmetic geometry (G öttingen, 2006), edited by H. Darmon et al., Clay Math. Proc. 8, Amer. Math. Soc., Providence, RI, 2009.
  • B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi sums, Wiley, New York, 1998.
  • F. Campana, “Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques”, Manuscripta Math. 117:4 (2005), 429–461. http://www.zentralblatt-math.org/zmath/en/search/?an=1129.14051Zbl 1129.14051
  • H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, 2nd ed., Cambridge University Press, 2005.
  • E. Peyre, “Hauteurs et mesures de Tamagawa sur les variétés de Fano”, Duke Math. J. 79:1 (1995), 101–218.
  • B. Poonen, “The projective line minus three fractional points”, 2006, http://www-math.mit.edu/~poonen/slides/campana_s.pdf.
  • W. M. Schmidt, Analytische Methoden für Diophantische Gleichungen. Einführende Vorlesungen, DMV Seminar 5, Birkhäuser, Basel, 1984.
  • R. C. Vaughan, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997.