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Small zeros of quadratic forms outside a union of varieties


Authors: Wai Kiu Chan, Lenny Fukshansky and Glenn R. Henshaw
Journal: Trans. Amer. Math. Soc. 366 (2014), 5587-5612
MSC (2010): Primary 11G50, 11E12, 11E39
DOI: https://doi.org/10.1090/S0002-9947-2014-06235-1
Published electronically: April 1, 2014
MathSciNet review: 3240936
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Abstract: Let $ F$ be a quadratic form in $ N \geq 2$ variables defined on a vector space $ V \subseteq K^N$ over a global field $ K$, and $ \mathcal {Z} \subseteq K^N$ be a finite union of varieties defined by families of homogeneous polynomials over $ K$. We show that if $ V \setminus \mathcal {Z}$ contains a nontrivial zero of $ F$, then there exists a linearly independent collection of small-height zeros of $ F$ in $ V\setminus \mathcal {Z}$, where the height bound does not depend on the height of $ \mathcal {Z}$, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace $ W$ of the quadratic space $ (V,F)$ such that $ W$ is not contained in  $ \mathcal {Z}$. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit.


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Additional Information

Wai Kiu Chan
Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
Email: wkchan@wesleyan.edu

Lenny Fukshansky
Affiliation: Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, California 91711
Email: lenny@cmc.edu

Glenn R. Henshaw
Affiliation: Department of Mathematics, California State University at Channel Islands, Camarillo, California 93012
Email: ghenshaw5974@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06235-1
Keywords: Heights, quadratic forms, Siegel's lemma
Received by editor(s): March 1, 2013
Published electronically: April 1, 2014
Additional Notes: The second author was partially supported by a grant from the Simons Foundation (#208969) and by the NSA Young Investigator Grant #1210223.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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