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arXiv · 2609.35617

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

Giovanni Longobardi·Valentina Pepe
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Abstract

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.

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Giovanni Longobardi, Valentina Pepe. 2026-09-28. The classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$. https://arxiv.org/abs/2609.35617

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