On the solvability of systems of bilinear equations in finite fields

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 $l\le \left(\genfrac{}{}{0ex}{}{k}{2}\right)$ bilinear equations \[ B (\tmmathbf{a}_i, \tmmathbf{a}_j) = λ_{i j}, \tmmathbf{a}_i \in \mathcal{A}_i, i = 1, ..., k. \] We show that the system is solvable for any ${\lambda }_{ij}\in {\mathbb{F}}_q^{\ast }$, $1\le i,j\le k$, given that the restricted sets ${\mathcal{A}}_i$'s are sufficiently large.

to appear in Proceedings of the American Mathematical Society

References in corpus (5)

• An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite field geometries
• Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness
• Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture
• On kaleidoscopic pseudo-randomness of finite Euclidean graphs
• On a Furstenberg-Katznelson-Weiss type theorem over finite fields