On the solvability of systems of equations revisited
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).
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.03086
- arXiv:
- arXiv:2405.03086
- Bibcode:
- 2024arXiv240503086P
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics
- E-Print:
- 12 pages