Skip to main content
SpringerLink
Log in

Inhomogeneous minimum of indefinite quadratic forms in five variables of type (3,2) or (2,3): A conjecture of watson

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

LetQ(x,y,z,t,u) be a real indefinite quadratic form in five variables of type (3,2) or (2,3) and determinantD≠0. The given any real numbersx 0,y 0,z 0,t 0,u 0 we can find integersx,y,z,t,u, satisfying

$$|Q(x + x_0 ,y + y_0 ,z + z_0 ,t + t_0 ,u + u_0 )| \leqslant (\frac{1}{4}|D|)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 5}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$5$}}} .$$

All the cases when the sign of equality holds are also determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birch, B. J.: The inhomogeneous minimum of quadratic forms of signature zero. Acta Arithm. 4, 85–98 (1958).

    Google Scholar 

  2. Blaney, H.: Indefinite quadratic forms inn variables. J. London Math. Soc. 23, 153–160 (1948).

    Google Scholar 

  3. Davenport, H.: Non-homogeneous ternary quadratic forms. Acta Math. 80, 65–95 (1948).

    Google Scholar 

  4. Dumir, V. C.: Inhomogeneous minimum of indefinite quaternary quadratic forms. Proc. Camb. Phil. Soc. 63, 277–290 (1967).

    Google Scholar 

  5. Dumir, V. C.: Positive values of inhomogeneous quaternary quadratic forms (II). J. Austral. Math. Soc. 8, 287–303 (1968).

    Google Scholar 

  6. Hans-Gill, R. J., andM. Raka: Asymmetric inequality for nonhomogeneous ternary quadratic forms. Mh. Math. 87, 281–295 (1979).

    Google Scholar 

  7. Jackson, T. H.: Small positive values of indefinite quadratic forms. J. London Math. Soc. (2) 1, 643–659 (1969).

    Google Scholar 

  8. Oppenheim, A.: Values of quadratic forms II. Quart J. Math. Oxford (2) 4, 60–66 (1953).

    Google Scholar 

  9. Oppenheim, A.: One sided inequalities for quadratic forms. II. Quaternary forms. Proc. London Math. Soc. 3, 417–429 (1953).

    Google Scholar 

  10. Watson, G. L.: Indefinite quadratic polynomials. Mathematika 7, 141–144 (1960).

    Google Scholar 

  11. Watson, G. L.: Indefinite quadratic forms in many variables: The inhomogeneous minimum and a generalization. Proc. London Math. Soc. (3) 12, 564–576 (1962).

    Google Scholar 

  12. Watson, G. L.: Asymmetric inequalities for indefinite quadratic forms. Proc. London Math. Soc. 18, 95–113 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Advanced Study in Mathematics, Panjab University, 160014, Chandigarh, India

    R. J. Hans-Gill & Madhu Raka

Authors
  1. R. J. Hans-Gill
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Madhu Raka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor S. Chowla on his Seventieth Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hans-Gill, R.J., Raka, M. Inhomogeneous minimum of indefinite quadratic forms in five variables of type (3,2) or (2,3): A conjecture of watson. Monatshefte für Mathematik 88, 305–320 (1979). https://doi.org/10.1007/BF01534249

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01534249

Keywords

Search

Navigation