Scattered linear sets in a finite projective line and translation planes

Abstract

In [15], Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in PG(1,qt). In this paper a similar construction of a translation plane Af obtained from any scattered linearized polynomial f(x) in Fqt[x] is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by Uf the Fq-subspace of Fqt2 associated with f(x). If f(x) and f′(x) are scattered, then Af and Af′ are isomorphic if and only if Uf and Uf′ belong to the same orbit under the action of ΓL(2,qt). 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,qt) there are cΓ(L) pairwise non-isomorphic translation planes, where cΓ(L) denotes the ΓL-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].