On the solvability of systems of equations revisited

TL;DR

This paper presents a new approach to analyze the solvability of systems of equations generated by bilinear forms, establishing lower bounds on the number of solutions for large sets in finite fields.

Contribution

It introduces a direct method to study bilinear form systems and improves previous bounds on the number of solutions in finite field settings.

Findings

For sets with size much greater than q^{5/3}, the number of solution triples is at least proportional to q^3.

The approach significantly improves previous results from 2009.

Provides a lower bound on the number of solutions for large sets in finite fields.

Abstract

In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot, \cdot)$ be a non-degenerate bilinear form and $E$ be a set in $\mathbb{F}_q^2$. We prove that if $|E|\gg q^{5/3}$ then the number of triples $(B(x, y), B(y, z), B(z, x))$ with $x, y, z\in E$ is at least $cq^3$ for some positive constant $c$. This significantly improves a result due to the fifth listed author (2009).