Skip to content
DraconiticResearch
Search Sources API
197,785 studies31 sourcesUpdated 52 min ago
PreprintarXiv licencearXiv

The classification of maximum scattered linear sets of PG(1,q5)

  • Giovanni Longobardi
  • Valentina Pepe
Open at sourcePDFDOICite

Abstract

We classify the maximum scattered 𝔽q-linear sets of PG(1,q5), proving that every such set is of pseudoregulus type or of Lunardon-Polverino type. Building on the reduction obtained by Lia, Longobardi and Zanella in [S. Lia, G. Longobardi and C. Zanella, Towards the classification of maximum scattered linear sets of PG(1,q5), Algebraic Combinatorics 9 (2026), 327-355], we show that the two remaining candidate families contain no scattered linear sets. Our approach reduces these candidates to two normal forms and establishes their non-scatteredness through the existence of rational points on associated algebraic varieties.

Related

  1. Maximum 2-scattered subspaces of V(r,q6) for every r≥ 3
    Daniele Bartoli, Francesco Ghiandoni, Alessandro Giannoni +2
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. A probabilistic construction of maximum h-scattered subspaces
    Alessandro Giannoni, Alessandro Neri
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds
    Javier Rodrigo, Sagrario Lantarón, María Dolores López
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. Shape optimisation of the maximum of the gradient of the torsion function on convex sets
    Michiel van den Berg, Dorin Bucur, Mickael Nahon
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Restricted Maximum Likelihood Estimation in Generalized Linear Mixed Models
    Luca Maestrini, Francis K. C. Hui, A. H. Welsh
    Statistical SciencePeer-reviewedLicence not statedFull text21 citedStatisticsOpenAlex

Cite

BibTeXRIS
BibTeX
@misc{longobardi2026classification,
  title = {{The classification of maximum scattered linear sets of $\mathrmPG(1,q^5)$}},
  author = {Giovanni Longobardi and Valentina Pepe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.35617},
  eprint = {2609.35617},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35617},
}
RIS
TY  - UNPB
TI  - The classification of maximum scattered linear sets of PG(1,q^5)
AU  - Giovanni Longobardi
AU  - Valentina Pepe
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.35617
UR  - https://arxiv.org/abs/2609.35617
AB  - We classify the maximum scattered 𝔽_q-linear sets of PG(1,q^5), proving that every such set is of pseudoregulus type or of Lunardon-Polverino type. Building on the reduction obtained by Lia, Longobardi and Zanella in [S. Lia, G. Longobardi and C. Zanella, Towards the classification of maximum scattered linear sets of PG(1,q^5), Algebraic Combinatorics 9 (2026), 327-355], we show that the two remaining candidate families contain no scattered linear sets. Our approach reduces these candidates to two normal forms and establishes their non-scatteredness through the existence of rational points on associated algebraic varieties.
ER  -