Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Small zeros of quadratic forms over number fields


Author: Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 302 (1987), 281-296
MSC: Primary 11E12
MathSciNet review: 887510
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Abstract: Let $ F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $ k$ and let $ A$ be a $ K \times N$ matrix over $ k$. We show that if the simultaneous equations $ F({\mathbf{x}}) = 0$ and $ A{\mathbf{x}} = 0$ hold on a subspace $ \mathfrak{X}$ of dimension $ L$ and $ L$ is maximal, then such a subspace $ \mathfrak{X}$ can be found with the height of $ \mathfrak{X}$ relatively small. In particular, the height of $ \mathfrak{X}$ can be explicitly bounded by an expression depending on the height of $ F$ and the height of $ A$. We use methods from geometry of numbers over adèle spaces and local to global techniques which generalize recent work of H. P. Schlickewei.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0887510-6
Article copyright: © Copyright 1987 American Mathematical Society