On the solvability of systems of bilinear equations in finite fields

TL;DR

This paper investigates the conditions under which systems of bilinear equations over finite fields are solvable, establishing that sufficiently large sets ensure solutions exist for any non-zero parameters.

Contribution

It provides new criteria linking the size of sets in finite fields to the solvability of bilinear systems with arbitrary non-zero constants.

Findings

Systems are solvable for any non-zero parameters if the sets are large enough.

The results extend understanding of bilinear equations over finite fields.

Conditions depend on the size of the sets relative to the field.

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 \[ B (\tmmathbf{a}_i, \tmmathbf{a}_j) = \lambda_{i j}, \tmmathbf{a}_i \in \mathcal{A}_i, i = 1, ..., k. \] 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.