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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, M5S1 A1, Toronto, Canada

    J. H. H. Chalk

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  1. J. H. H. Chalk
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Chalk, J.H.H. Linearly independent zeros of quadratic forms over number-fields. Monatshefte für Mathematik 90, 13–25 (1980). https://doi.org/10.1007/BF01641708

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  • DOI: https://doi.org/10.1007/BF01641708

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