Small zeros of quadratic forms outside a union of varieties
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- by Wai Kiu Chan, Lenny Fukshansky and Glenn R. Henshaw PDF
- Trans. Amer. Math. Soc. 366 (2014), 5587-5612 Request permission
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.References
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Additional Information
- Wai Kiu Chan
- Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
- MR Author ID: 336822
- Email: wkchan@wesleyan.edu
- Lenny Fukshansky
- Affiliation: Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, California 91711
- MR Author ID: 740792
- Email: lenny@cmc.edu
- Glenn R. Henshaw
- Affiliation: Department of Mathematics, California State University at Channel Islands, Camarillo, California 93012
- Email: ghenshaw5974@gmail.com
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3240936