Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$

TL;DR

This paper investigates a class of scattered linear sets called pseudoregulus type in projective spaces, generalizing previous results and characterizing certain semifields through these linear sets.

Contribution

It introduces and studies maximum scattered linear sets of pseudoregulus type in higher-dimensional projective spaces, extending prior work and linking to semifield classifications.

Findings

Characterization of linear sets of pseudoregulus type in PG(2n-1,q^t)

Generalization of results from PG(3,q^3) to higher dimensions

Identification of connections between linear sets and classical semifields

Abstract

In this paper we study a family of scattered $\F_q$--linear sets of rank $tn$ of the projective space $PG(2n-1,q^t)$ ($n \geq 1$, $t\geq 3$), called of {\it pseudoregulus type}, generalizing results contained in [G. Marino, O. Polverino, R. Trombetti: On ${\mathbb F}_q$--linear sets of $\PG(3,q^3)$ and semifields, {\em J. Combin. Theory Ser. A}, {\bf 114} (2007), 769--788] and in [M. Lavrauw, G. Van de Voorde: Scattered linear sets and pseudoreguli, preprint]. As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twisted Fields and the 2-dimensional Knuth semifields.