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The weak cylinder conjecture for polynomial graphs of bounded degree

  • Daniele Bartoli
  • Alessandro Giannoni
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Abstract

Ball's weak cylinder conjecture asserts that a set of p2 points in AG(3,p) with at least p non-determined directions is a union of p parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most D, provided that p>D9/2. More precisely, a non-cylindrical graph of degree D≥3 satisfies |𝒩S|≤ p+1−⌈ p/(D−1)⌉ under the same hypothesis. This bound is attained when p≡1pmodD−1. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in p. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

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Cite

BibTeXRIS
BibTeX
@misc{bartoli2026weak,
  title = {{The weak cylinder conjecture for polynomial graphs of bounded degree}},
  author = {Daniele Bartoli and Alessandro Giannoni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16803},
  eprint = {2610.16803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16803},
}
RIS
TY  - UNPB
TI  - The weak cylinder conjecture for polynomial graphs of bounded degree
AU  - Daniele Bartoli
AU  - Alessandro Giannoni
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16803
UR  - https://arxiv.org/abs/2610.16803
AB  - Ball's weak cylinder conjecture asserts that a set of p^2 points in AG(3,p) with at least p non-determined directions is a union of p parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most D, provided that p>D^9/2. More precisely, a non-cylindrical graph of degree D≥3 satisfies |𝒩_S|≤ p+1−⌈ p/(D−1)⌉ under the same hypothesis. This bound is attained when p≡1pmodD−1. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in p. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.
ER  -