Monatshefte für Mathematik

, Volume 90, Issue 1, pp 13–25

Linearly independent zeros of quadratic forms over number-fields

  • J. H. H. Chalk
Article

DOI: 10.1007/BF01641708

Cite this article as:
Chalk, J.H.H. Monatshefte für Mathematik (1980) 90: 13. doi:10.1007/BF01641708

Abstract

LetK be an algebraic number-field of degree [K:Q] =n ⩾ 1 and letO denote some fixed order ofK. Let
, be a quadratic form which represents zero for some
. For the special caseK =Q,O =Z, theorems ofCassels and ofDavenport provide estimates for the magnitude (in terms of the coefficients off(x)) of a zero and of a pair of linearly independent zeros off, respectively. Recently,Raghavan extendedCassels' result to arbitraryK. In this article, a new proof ofDavenport's theorem for a pair of linearly independent zeros is given which not only provides explicit constants in the estimates but also extends to generalK. A refinement of this proof leads to effectively computable bounds for rational representations of a numbern≠0 byf.

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • J. H. H. Chalk
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada