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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.09910 |
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| _version_ | 1866911376970088448 |
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| author | Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor |
| author_facet | Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor |
| contents | A set of points $S \subseteq \mathbb{F}_p^n$ is called \emph{$p$-divisible} if every affine hyperplane in $\mathbb{F}_p^n$ intersects $S$ in $0 \pmod p$ points. The Strong Cylinder Conjecture of Ball asserts that if
$S$ is a $p$-divisible set of $p^2$ points in $\mathbb{F}_p^3$, then $S$ is a cylinder. In this paper, we show that every $p$-divisible multiset $S$ is both a $\mathbb{F}_p$-linear and $\mathbb{Z}$-linear combination of characteristic functions of cylinders. In addition, the multisets of size $p^2$ are $\Z$-linear combinations of a plane and weighted differences of parallel lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09910 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$ Kiss, Gergely Markó, Ádám Nagy, Zoltán Lóránt Somlai, Gábor Combinatorics A set of points $S \subseteq \mathbb{F}_p^n$ is called \emph{$p$-divisible} if every affine hyperplane in $\mathbb{F}_p^n$ intersects $S$ in $0 \pmod p$ points. The Strong Cylinder Conjecture of Ball asserts that if $S$ is a $p$-divisible set of $p^2$ points in $\mathbb{F}_p^3$, then $S$ is a cylinder. In this paper, we show that every $p$-divisible multiset $S$ is both a $\mathbb{F}_p$-linear and $\mathbb{Z}$-linear combination of characteristic functions of cylinders. In addition, the multisets of size $p^2$ are $\Z$-linear combinations of a plane and weighted differences of parallel lines. |
| title | Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.09910 |