Scattered linear sets in a finite projective line and translation planes

In [15], Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $PG(1,q^t)$. In this paper a similar construction of a translation plane $A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $F_q$-subspace of $T_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $A_f$ and $A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$ belong to the same orbit under the action of $\GaL(2,q^t)$. This gives rise to the same number of distinct translation planes as the number of inequivalent scattered linearized polynomials. In particular, for any scattered linear set $L$ of maximum rank in $PG(1,q^t)$ there are $c_\Gamma(L)$ pairwise non-isomorphic translation planes, where $c_\Gamma(L)$ denotes the $\GaL$-class of $L$, as defined in [5] by Csajbók, Marino and Polverino. A result by Jha and Johnson [8] allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined in [15].