On the solvability of systems of equations revisited

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 $\mathrm{\mid }E\mathrm{\mid }\gg q^{5\mathrm{/}3}$ then the number of triples $(B(x,y),B(y,z),B(z,x))$ with $x,y,z\in E$ is at least $c{q}^3$ for some positive constant $c$. This significantly improves a result due to the fifth listed author (2009).