Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

TL;DR

This work classifies maximum scattered ${\mathbb F}_q$-linear sets in ${\mathrm{PG}}(1,q^5)$ via projecting configurations from a canonical ${\mathbb F}_q$-subgeometry $\Sigma\subset{\mathrm{PG}}(4,q^5)$ through a vertex plane $\Gamma$. By analyzing the mutual positions of $\Gamma$ and its conjugates under a cyclic group $\mathbb G$ of order 5, the authors show that if at least one of $A=\Gamma\cap\Gamma^{\sigma^4}$ or $B=\Gamma\cap\Gamma^{\sigma^3}$ has rank 5, then the MSLS must be LP type; when $\operatorname{rk}A=\operatorname{rk}B=4$ they derive two candidate polynomial forms for a potential new type, but extensive computer checks for $q\le 25$ find no new MSLS. The main conclusion is that, for rank-5 vertices, no new MSLS exist beyond pseudoregulus and LP types, and the remaining rank-4 case yields no new examples within explored parameters, narrowing the classification for $\mathrm{PG}(1,q^5)$. The approach translates geometric conditions into algebraic curves, enabling the use of the Hasse–Weil theorem and computational verification to rule out new instances in substantial parameter ranges.

Abstract

Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $Σ$ of $\mathrm{PG}(4,q^5)$ from a plane $Γ$ external to the secant variety to $Σ$. The pair $(Γ,Σ)$ will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in $\mathrm{PG}(1,q^5)$, namely those of pseudoregulus and LP type, have been characterized in the literature by B. Csajbók, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. Let $(Γ,Σ)$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$, let $σ$ be a generator of $\mathbb{G}=\mathrm{P}Γ\mathrm{L}(5,q^5)_Σ$, and $A=Γ\capΓ^{σ^4}$, $B=Γ\capΓ^{σ^3}$. If $A$ and $B$ are not both points, then the projected linear set is of pseudoregulus type. Then, suppose that they are points. The rank of a point $X$ is the vectorial dimension of the span of the orbit of $X$ under the action of $\mathbb{G}$. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points $A$ and $B$ has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that $\mathrm{rk} A=\mathrm{rk} B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\leq 25$, no new maximum scattered linear set exists.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Proposition 2.2
• proof