# On the solvability of systems of bilinear equations in finite fields

@article{Vinh2009OnTS,
title={On the solvability of systems of bilinear equations in finite fields},
author={L. Vinh},
journal={arXiv: Combinatorics},
year={2009}
}
• L. Vinh
• Published 2009
• Mathematics
• arXiv: Combinatorics
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 $B (\tmmathbf{a}_i, \tmmathbf{a}_j) = \lambda_{i j}, \tmmathbf{a}_i \in \mathcal{A}_i, i = 1, ..., k.$ 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 $\mathcal{A}_i$'s are sufficiently large.
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