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arXiv · 2405.03086

On the solvability of systems of equations revisited

Thang Pham·Steven Senger·Nguyen Trung-Tuan·Nguyen Duc-Thang·Le Anh Vinh
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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).

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Thang Pham, Steven Senger, Nguyen Trung-Tuan, Nguyen Duc-Thang, Le Anh Vinh. 2024-05-06. On the solvability of systems of equations revisited. https://arxiv.org/abs/2405.03086

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