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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.17005 |
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| _version_ | 1866913357461716992 |
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| author | Dong, Dingding Li, Anqi Zhao, Yufei |
| author_facet | Dong, Dingding Li, Anqi Zhao, Yufei |
| contents | A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as $n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least as many monochromatic solutions to $L$ as a random 2-coloring. The notion of common linear systems is analogous to that of common graphs, i.e., graphs whose monochromatic density in 2-edge-coloring of cliques is asymptotically minimized by the random coloring. Saad and Wolf initiated a systematic study on identifying common linear systems, built upon the earlier work of Cameron-Cilleruelo-Serra. When $L$ is a single equation, Fox-Pham-Zhao gave a complete characterization of common linear equations. When $L$ consists of two equations, Kamčev-Liebenau-Morrison showed that irredundant $2\times 4$ linear systems are always uncommon. In this work, (1) we determine commonness of all $2\times 5$ linear systems up to a small number of cases, and (2) we show that all $2\times k$ linear systems with $k$ even and girth (minimum number of nonzero coefficients of a nonzero equation spanned by the system) $k-1$ are uncommon, answering a question of Kamčev-Liebenau-Morrison. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17005 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uncommon linear systems of two equations Dong, Dingding Li, Anqi Zhao, Yufei Combinatorics Number Theory A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as $n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least as many monochromatic solutions to $L$ as a random 2-coloring. The notion of common linear systems is analogous to that of common graphs, i.e., graphs whose monochromatic density in 2-edge-coloring of cliques is asymptotically minimized by the random coloring. Saad and Wolf initiated a systematic study on identifying common linear systems, built upon the earlier work of Cameron-Cilleruelo-Serra. When $L$ is a single equation, Fox-Pham-Zhao gave a complete characterization of common linear equations. When $L$ consists of two equations, Kamčev-Liebenau-Morrison showed that irredundant $2\times 4$ linear systems are always uncommon. In this work, (1) we determine commonness of all $2\times 5$ linear systems up to a small number of cases, and (2) we show that all $2\times k$ linear systems with $k$ even and girth (minimum number of nonzero coefficients of a nonzero equation spanned by the system) $k-1$ are uncommon, answering a question of Kamčev-Liebenau-Morrison. |
| title | Uncommon linear systems of two equations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2404.17005 |