Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-29T19:07:02.014Z Has data issue: false hasContentIssue false
  • English
  • Français

On certain cubic forms in seven variables

Published online by Cambridge University Press:  24 March 2010

M. P. HARVEY
M. P. HARVEY*
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The main result of this paper is a proof of the expected asymptotic formula for the density of zeros of a family of cubic forms in seven variables. This is established using the Hardy–Littlewood circle method.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 1 , July 2010 , pp. 21 - 47
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baker, R. C.Diagonal cubic equations. II. Acta Arith. 53 (1989), no. 3, 217–250.CrossRefGoogle Scholar
[2]Birch, B. J., Davenport, H. and Lewis, D. J.The addition of norm forms. Mathematika 9(1962), 75–82.CrossRefGoogle Scholar
[3]Boklan, K. D.A reduction technique in Waring's problem. I. Acta Arith. 65 (1993), no. 2, 147–161.CrossRefGoogle Scholar
[4]Browning, T. D. Rational points on cubic hypersurfaces that split off a form, submitted.Google Scholar
[5]Brüdern, J. and Wooley, T. D.The addition of binary cubic forms. Royal Soc. London Phil. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1738, 701–737.CrossRefGoogle Scholar
[6]Chowla, S. and Davenport, H.On Weyl's inequality and Waring's problem for cubes. Acta Arith. 6 (1960/1961), 505–521.CrossRefGoogle Scholar
[7]Colliot–Thélène, J. L. and Salberger, P.Arithmetic on some singular cubic hypersurfaces. Proc. London Math. Soc. (3) 58 (1989), no. 3, 519–549.Google Scholar
[8]Davenport, H.Cubic forms in thirty-two variables. Phil. Trans. Royal Soc. London. Ser. A 251 (1959), 193–232.Google Scholar
[9]Davenport, H. Analytic Methods for Diophantine Equations and Diophantine Inequalities. 2nd ed. edited and prepared for publication by Browning, T. D., Cambridge Mathematical Library (Cambridge University Press, 2005).CrossRefGoogle Scholar
[10]Davenport, H.Cubic forms in sixteen variables. Proc. Royal Soc. Ser. A 272 (1963), 285–303.Google Scholar
[11]Franke, J., Manin, Y. I. and Tschinkel, Y.Rational points of bounded height on Fano varieties. Invent. Math. 95 (1989), no. 2, 421–435.CrossRefGoogle Scholar
[12]Heath–Brown, D. R.Cubic forms in 14 variables. Invent. Math. 170 (2007), no. 1, 199–230.CrossRefGoogle Scholar
[13]Heilbronn, H. Zeta functions and L-functions. Algebraic Number Theory (Cassels, J. W. S. and Fröhlich, A., Eds.), (Academic Press, 1977), pp. 204–230.Google Scholar
[14]Hooley, C.On nonary cubic forms. J. Reine Angew. Math. 386 (1988), 32–98.Google Scholar
[15]Hooley, C.On the number of points on a complete intersection over a finite field. J. Number Theory 38 (1991), 338–358.CrossRefGoogle Scholar
[16]Ireland, K. and Rosen, M. A.A classical introduction to modern number theory, second edition. Graduate Texts in Mathematics 84 (Springer-Verlag, 1990).CrossRefGoogle Scholar
[17]Iwaniec, H. and Kowalski, E.Analytic number theory. Amer. Math. Soc. Colloq. Publ. 53 (2004).Google Scholar
[18]Moroz, B. Z.Analytic arithmetic in algebraic number fields, Lecture Notes in Mathematics, 1205 (Springer-Verlag, 1986).Google Scholar
[19]Vaughan, R. C.On Waring's problem for cubes. J. Reine Angew. Math. 365 (1986), 122–170.Google Scholar
[20]Vaughan, R. C.A new iterative method in Waring's problem. Acta Math. 162 (1989), no. 1-2, 1–71.CrossRefGoogle Scholar
[21]Vaughan, R. C.The Hardy-Littlewood Method, 2nd edition (Cambridge University Press, 1997).CrossRefGoogle Scholar
[22]Wolke, D.Multiplikative funktionen auf schnell wachsenden folgen (German). J. Reine Angew. Math. 251 (1971), 54–67.Google Scholar