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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Small zeros of quadratic forms over number fields. II


Author: Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 313 (1989), 671-686
MSC: Primary 11E12; Secondary 11H55
MathSciNet review: 940914
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Abstract: Let $ F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $ k$ and let $ \mathcal{Z}$ be a subspace of $ {k^N}$ of dimension $ M,1 \leq M \leq N$. If $ F$ restricted to $ \mathcal{Z}$ vanishes on a subspace of dimension $ L,1 \leq L < M$, and if the rank of $ F$ restricted to $ \mathcal{Z}$ is greater than $ M - L$, then we show that $ F$ must vanish on $ M - L + 1$ distinct subspaces $ {\mathcal{X}_0},{\mathcal{X}_1}, \ldots ,{\mathcal{X}_{M - L}}$ in $ \mathcal{Z}$ each of which has dimension $ L$. Moreover, we show that for each pair $ {\mathcal{X}_0},{\mathcal{X}_1},1 \leq l \leq M - L$, the product of their heights $ H({\mathcal{X}_0})H({\mathcal{X}_1})$ is relatively small. Our results generalize recent work of Schlickewei and Schmidt.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0940914-7
PII: S 0002-9947(1989)0940914-7
Article copyright: © Copyright 1989 American Mathematical Society