• D. R. Heath-Brown
Conference paper
Part of the NATO Science Series book series (NAII, volume 237)


The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the Hardy– Littlewood circle method. There are a number of good texts available on the circle method, but the reader may particularly wish to study the books (Davenport, 2005) and (Vaughan, 1997).


Rational Point Asymptotic Formula Algebraic Variety Integer Point Integer Vector 
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  1. Birch, B. J. (1961/1962) Forms in many variables, Proc. Roy. Soc. Ser. A 265, 245–263.MathSciNetGoogle Scholar
  2. Cook, R. J. (1971) Simultaneous quadratic equations, J. London Math. Soc. (2) 4, 319–326.MATHCrossRefMathSciNetGoogle Scholar
  3. Davenport, H. (2005) Analytic methods for Diophantine equations and Diophantine inequalities, In Cambridge Mathematical Library, Cambridge, Cambridge University Press.Google Scholar
  4. Davenport, H. and Lewis, D. J. (1969) Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264, 557–595.MATHMathSciNetGoogle Scholar
  5. Deligne, P. (1980) La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 52, 137–252.MATHMathSciNetGoogle Scholar
  6. Ford, K. (1995) New estimates for mean values of Weyl sums, Internat. Math. Res. Notices pp. 155–171.Google Scholar
  7. Heath-Brown, D. R. (1996) A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481, 149–206.MATHMathSciNetGoogle Scholar
  8. Heath-Brown, D. R. (2003) Linear relations amongst sums of two squares, In London Math. Soc. Lecture Note Ser., Vol. 303 of London Math. Soc. Lecture Note Ser., Cambridge, pp. 133–176, Cambridge Univ. Press.Google Scholar
  9. Heath-Brown, D. R. and Skorobogatov, A. N. (2002) Rational solutions of certain equations involving norms, Acta Math. 189, 161–177.MATHCrossRefMathSciNetGoogle Scholar
  10. Hooley, C. (1994) On nonary cubic forms. III, J. Reine Angew. Math. 456, 53–63.MATHMathSciNetGoogle Scholar
  11. Vaughan, R. C. (1986) On Waring’s problem for cubes, J. Reine Angew. Math 365, 122–170.MATHMathSciNetGoogle Scholar
  12. Vaughan, R. C. (1997) The Hardy—Littlewood method, In Cambridge Tracts in Mathematics, Vol. 125, Cambridge-New York, Cambridge University Press.Google Scholar
  13. Wang, Y. (1991) Diophantine equations and inequalities in algebraic number fields, Berlin, Springer-Verlag.MATHGoogle Scholar
  14. Weyl, H. (1916) Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77, 313–352.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • D. R. Heath-Brown
    • 1
  1. 1.Oxford UniversityOxford

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