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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Small zeros of quadratic forms mod $ P^2$


Authors: Todd Cochrane and Ali H. Hakami
Journal: Proc. Amer. Math. Soc. 140 (2012), 4041-4052
MSC (2010): Primary 11D79, 11E08, 11H50, 11H55
Published electronically: March 30, 2012
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Abstract: Let $ Q(\mathbf x)$ be a quadratic form over $ \mathbb{Z}$ in $ n$ variables, $ p$ be an odd prime and $ \Vert \mathbf x\Vert= \max _i \vert x_i\vert$. A solution of the congruence $ Q(\mathbf x) \equiv 0 \pmod {p^2}$ is said to be nontrivial if $ p \nmid x_i$ for some $ i$. We prove that if this congruence has a nontrivial solution, then it has a nontrivial solution with $ \Vert\mathbf x\Vert\le p$. We also give estimates on the number of small nontrivial solutions of the congruence and show that there exists a set of $ n$ linearly independent nontrivial solutions of size $ \Vert\mathbf x\Vert \le (2^{n+1}+1)p$, provided that $ n \ge 4$ is even and $ Q(\mathbf x)$ is nonsingular $ \pmod p$.


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Additional Information

Todd Cochrane
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: cochrane@math.ksu.edu

Ali H. Hakami
Affiliation: Department of Mathematics, King Khalid University, Abha, Saudi Arabia 61431
Email: aalhakami@kku.edu.sa

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11310-3
PII: S 0002-9939(2012)11310-3
Keywords: Quadratic forms, congruences, small solutions
Received by editor(s): January 26, 2011
Received by editor(s) in revised form: May 17, 2011
Published electronically: March 30, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.