Scattered linear sets in a finite projective line and translation planes

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.