On the solvability of systems of equations revisited
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909190753091584 |
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| author | Pham, Thang Senger, Steven Trung-Tuan, Nguyen Duc-Thang, Nguyen Vinh, Le Anh |
| author_facet | Pham, Thang Senger, Steven Trung-Tuan, Nguyen Duc-Thang, Nguyen Vinh, Le Anh |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the solvability of systems of equations revisited Pham, Thang Senger, Steven Trung-Tuan, Nguyen Duc-Thang, Nguyen Vinh, Le Anh Number Theory Combinatorics 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). |
| title | On the solvability of systems of equations revisited |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2405.03086 |