, Volume 90, Issue 1, pp 13-25

Linearly independent zeros of quadratic forms over number-fields

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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.