AI 中文总结
本文分类了 $\mathrm{PG}(1,q^5)$ 中的最大散在 $\mathbb{F}_q$-线性集，证明其仅为伪正则型或 Lunardon-Polverino 型，通过约化与代数簇有理点方法排除了剩余候选族。
AI 中文摘要
我们分类了 $\mathrm{PG}(1,q^5)$ 中的最大散在 $\mathbb{F}_q$-线性集，证明了每个这样的集合都是伪正则型或 Lunardon-Polverino 型。基于 Lia、Longobardi 和 Zanella 在 [S. Lia, G. Longobardi and C. Zanella, Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$, Algebraic Combinatorics 9 (2026), 327-355] 中得到的约化结果，我们证明了剩余的两个候选族不包含散在线性集。我们的方法将这两个候选族化为两种正规形式，并通过相关代数簇上有理点的存在性确立了它们的非散在性。
英文摘要
We classify the maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^5)$, 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 $\mathrm{PG}(1,q^5)$, 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.
