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

Source 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.