Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$

TL;DR

This paper investigates the structure of $p$-divisible point sets in three-dimensional finite fields, showing they can be expressed as linear combinations of cylinders and other geometric objects, advancing understanding of the Strong Cylinder Conjecture.

Contribution

It proves that $p$-divisible multisets in $_p^3$ are linear combinations of cylinders and simpler geometric configurations, providing new structural insights.

Findings

Every $p$-divisible multiset is a linear combination of cylinders.

Multisets of size $p^2$ are combinations of a plane and differences of parallel lines.

The results support the structure predicted by the Strong Cylinder Conjecture.

Abstract

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.