Scattered linear sets in a finite projective line and translation planes

TL;DR

This paper generalizes the construction of translation planes from scattered linear sets in projective lines, introducing new classes derived from scattered linearized polynomials and analyzing their automorphisms.

Contribution

It extends previous work by constructing translation planes from any scattered linearized polynomial, classifying them via group orbits, and describing their automorphism groups.

Findings

Constructed new classes of translation planes from scattered linearized polynomials.

Established criteria for isomorphism of these planes based on group orbits.

Described automorphism groups for planes from non-pseudoregulus type linear sets.

Abstract

Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$ belong to the same orbit under the action of $\Gamma\mathrm L(2,q^t)$. This gives rise to as many distinct translation planes as there are inequivalent scattered linearized polynomials. In particular, for any scattered linear set $L$ of maximum rank in $\mathrm{PG}(1,q^t)$ there are $c_\Gamma(L)$ pairwise non-isomorphic translation planes, where $c_\Gamma(L)$ denotes the $\Gamma\mathrm L$-class of $L$, as defined by Csajb\'ok, Marino and Polverino. A result by Jha and Johnson allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined by Lunardon and Polverino.