On the solvability of systems of bilinear equations in finite fields
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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 \[ B (\mathbfit {a}_i, \mathbfit {a}_j) = \lambda _{i j}, \mathbfit {a}_i \in \mathcal {A}_i, i = 1, \ldots , k. \] 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.References
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Additional Information
- Le Anh Vinh
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 798264
- Email: vinh@math.harvard.edu
- Received by editor(s): December 1, 2008
- Published electronically: May 4, 2009
- Communicated by: Ken Ono
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2889-2898
- MSC (2000): Primary 11L40, 11T30; Secondary 11E39
- DOI: https://doi.org/10.1090/S0002-9939-09-09947-X
- MathSciNet review: 2506446