On the solvability of systems of bilinear equations in finite fields

Vinh, Le

doi:10.1090/S0002-9939-09-09947-X

On the solvability of systems of bilinear equations in finite fields
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Proc. Amer. Math. Soc. 137 (2009), 2889-2898 Request permission

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.

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