Small zeros of quadratic forms mod $P^2$
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- by Todd Cochrane and Ali H. Hakami PDF
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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 $\| \mathbf x\|= \max _i |x_i|$. 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 $\|\mathbf x\|\le p$. 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 $\|\mathbf x\| \le (2^{n+1}+1)p$, provided that $n \ge 4$ is even and $Q(\mathbf x)$ is nonsingular $\pmod p$.References
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Additional Information
- Todd Cochrane
- Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
- MR Author ID: 227122
- Email: cochrane@math.ksu.edu
- Ali H. Hakami
- Affiliation: Department of Mathematics, King Khalid University, Abha, Saudi Arabia 61431
- Email: aalhakami@kku.edu.sa
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4041-4052
- MSC (2010): Primary 11D79, 11E08, 11H50, 11H55
- DOI: https://doi.org/10.1090/S0002-9939-2012-11310-3
- MathSciNet review: 2957194