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 (·, ·) be a non-degenerate bilinear form and E be a set in Fq². We prove that if |E| q5/3 then the number of triples $(B(x, y), B(y, z), B(z, x))$ with x, y, z∈ E is at least cq³ for some positive constant c. This significantly improves a result due to the fifth listed author (2009).