A geometric characterization of known maximum scattered linear sets of $\mathrm{PG}(1,q^n)$
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929333902245888 |
|---|---|
| author | Grimaldi, Giovanni Giuseppe Gupta, Somi Longobardi, Giovanni Trombetti, Rocco |
| author_facet | Grimaldi, Giovanni Giuseppe Gupta, Somi Longobardi, Giovanni Trombetti, Rocco |
| contents | An $\mathbb{F}_q$- linear set $L=L_U$ of $Λ=\mathrm{PG}(V, \mathbb{F}_{q^n}) \cong \mathrm{PG}(r-1,q^n)$ is a set of points defined by non-zero vectors of an $\mathbb{F}_q$-subspace $U$ of $V$. The integer $\dim_{\mathbb{F}_q} U$ is called the rank of $L$. In [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004)], it was proven that any $\mathbb{F}_q$-linear set $L$ of $Λ$ of rank $u$ such that $\langle L \rangle=Λ$ is either a canonical subgeometry of $Λ$ or there are a $(u-r-1)$-dimensional subspace $Γ$ of $\mathrm{PG}(u-1,q^n) \supset Λ$ disjoint from $Λ$ and a canonical subgeometry $Σ\cong \mathrm{PG}(u-1,q)$ disjoint from $Γ$ such that $L$ is the projection of $Σ$ from $Γ$ onto $Λ$. The subspace $Γ$ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex $Γ$ for a peculiar class of linear sets of rank $u = n(r - 1)$ in $\mathrm{PG}(r - 1, q^n)$ called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line $\mathrm{PG}(1,q^n)$ introduced from 2018 onward, by means of certain properties of their projection vertices, as done in [B. Csajbók, C. Zanella: On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1, q^t)$, Finite Fields Appl. 41 (2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1, q^n)$. Discrete Math. 343(5) (2020)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric characterization of known maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ Grimaldi, Giovanni Giuseppe Gupta, Somi Longobardi, Giovanni Trombetti, Rocco Combinatorics 51E20, 05B25 An $\mathbb{F}_q$- linear set $L=L_U$ of $Λ=\mathrm{PG}(V, \mathbb{F}_{q^n}) \cong \mathrm{PG}(r-1,q^n)$ is a set of points defined by non-zero vectors of an $\mathbb{F}_q$-subspace $U$ of $V$. The integer $\dim_{\mathbb{F}_q} U$ is called the rank of $L$. In [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004)], it was proven that any $\mathbb{F}_q$-linear set $L$ of $Λ$ of rank $u$ such that $\langle L \rangle=Λ$ is either a canonical subgeometry of $Λ$ or there are a $(u-r-1)$-dimensional subspace $Γ$ of $\mathrm{PG}(u-1,q^n) \supset Λ$ disjoint from $Λ$ and a canonical subgeometry $Σ\cong \mathrm{PG}(u-1,q)$ disjoint from $Γ$ such that $L$ is the projection of $Σ$ from $Γ$ onto $Λ$. The subspace $Γ$ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex $Γ$ for a peculiar class of linear sets of rank $u = n(r - 1)$ in $\mathrm{PG}(r - 1, q^n)$ called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line $\mathrm{PG}(1,q^n)$ introduced from 2018 onward, by means of certain properties of their projection vertices, as done in [B. Csajbók, C. Zanella: On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1, q^t)$, Finite Fields Appl. 41 (2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1, q^n)$. Discrete Math. 343(5) (2020)]. |
| title | A geometric characterization of known maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ |
| topic | Combinatorics 51E20, 05B25 |
| url | https://arxiv.org/abs/2405.01374 |