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

 

On the solvability of systems of bilinear equations in finite fields


Author: Le Anh Vinh
Journal: Proc. Amer. Math. Soc. 137 (2009), 2889-2898
MSC (2000): Primary 11L40, 11T30; Secondary 11E39
Published electronically: May 4, 2009
MathSciNet review: 2506446
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Abstract: Given $ k$ sets $ \mathcal{A}_i \subseteq \mathbb{F}_q^d$ and a non-degenerate bilinear form $ B$ in $ \mathbb{F}_q^d$, we consider the system of $ l \leq \binom{k}{2}$ bilinear equations

$\displaystyle B ( \ensuremath{\boldsymbol{a}}_i, \ensuremath{\boldsymbol{a}}_j... ...bda_{i j}, \ensuremath{\boldsymbol{a}}_i \in \mathcal{A}_i, i = 1, \ldots, k. $

We show that the system is solvable for any $ \lambda_{i j} \in \mathbb{F}_q^{*}$, $ 1 \leq i,j \leq k$, given that the restricted sets $ \mathcal{A}_i$ are sufficiently large.


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Additional Information

Le Anh Vinh
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: vinh@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09947-X
PII: S 0002-9939(09)09947-X
Keywords: Bilinear equations, finite fields
Received by editor(s): December 1, 2008
Published electronically: May 4, 2009
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.