Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T15:32:04.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Inhomogeneous minima of a class of ternary quadratic forms

Published online by Cambridge University Press:  09 April 2009

Madhu Raka
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let denote the kth successive inhomogeneous minima for positive values of real indefinite ternary quadratic forms of type (2, 1). Here it is proved that for the class of zero forms, All the critical forms have also been obtained. is already known. For non-zero forms it is proved that .

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 3 , December 1993 , pp. 334 - 354
Copyright
Copyright © Australian Mathematical Society 1993

References

Bambah, R. P., Dumir, V. C. and Hans-Gill, R. J. (1981), ‘Positive values of non-homogeneous indefinite quadratic forms, in: Topics in classical number theory, Colloq. Math. Soc. J´nes Bolyai, (North, Amsterdam), pp. 111–170.Google Scholar
Bambah, R. P., Dumir, V. C. and Hans-Gill, R. J. (1984), ‘Positive values of non-homogeneous indefinite quadratic forms II’, J. Number Theory 18, 313–341.CrossRefGoogle Scholar
Barnes, E. S. (1954), ‘The inhomogeneous minimum of a ternary quadratic forms’, Acta Math. 92, 13–33.CrossRefGoogle Scholar
Barnes, E. S. (1955), ‘The non negative values of quadratic forms’, Proc. London Math Soc. 5 (3), 185–196.CrossRefGoogle Scholar
Barnes, E. S. (1956), ‘The inhomogeneous minimum of a ternary quadratic forms (II)’, Acta Math. 96, 67–97.CrossRefGoogle Scholar
Barnes, E. S. (1961), ‘The positive values of inhomogeneous ternary quadratic forms’, J. Austral. Math. Soc. (Series A) 2, 127–132.CrossRefGoogle Scholar
Blaney, H. (1950a), ‘Indefinite ternary quadratic forms’, Quart.J. Math., Oxford Ser. 1 (2), 252–69.CrossRefGoogle Scholar
Blaney, H. (1950b), ‘Some asymmetric inequalities’, Math. Proc. Cambridge Philos. Soc. 46, 359–376.CrossRefGoogle Scholar
Davenport, H. (1946), ‘Non-homogeneous binary quadratic forms’, Proc. Kon. Nederl. Ak.ad. Wetensch. 49, 815–821.Google Scholar
Davenport, H. (1948), ‘Non-homogeneous ternary quadratic forms’, Acta Math. 80, 65–95.CrossRefGoogle Scholar
Dumir, V. C. (1967), ‘Asymmetric inequalities for non-homogeneous ternary quadratic forms’, Math. Proc. Cambridge Philos. Soc. 63, 291–303.CrossRefGoogle Scholar
Dumir, V. C. (1968), ‘Positive values of inhomogeneous quadratic forms I’, J. Austral. Math. Soc. (SeriesA) 8, 87–101.CrossRefGoogle Scholar
Dumir, V. C. and Hans-Gill, R. J. (1991), ‘The second minimum for positive values of non-homogeneous ternary quadratic forms of type (1, 2)’, unpublished.Google Scholar
Grover, V. K. and Raka, M. (1991), ‘On inhomogeneous minima of indefinite binary quadratic forms’, Acta Math. 167, 287–298.CrossRefGoogle Scholar
Macbeath, A. M. (1951), ‘A new sequence of minima in the geometry of numbers’, Math. Proc. Cambridge Philos. Soc. 47, 266–273.CrossRefGoogle Scholar
Margulis, G. A. (1987), ‘Indefinite quadratic forms and unipotent flows on homogeneous spaces’, Acad. Sci Paris Sér. I Math. 304, 249–253.Google Scholar
Raka, M. (1983), ‘On a conjecture of Watson’, Math. Proc. Cambridge Philos. Soc. 94, 9–22.CrossRefGoogle Scholar
Vulakh, I. Y. (1985), ‘On minima of rational indefinite quadratic forms’, J. Number Theory 21, 275–285.CrossRefGoogle Scholar