Towards the classification of maximum scattered linear sets of PG(1,q^5)

Every maximum scattered linear set in PG(1, q5) is the projection of an Fq-subgeometry Σ of PG(4, q5) from a plane Γ external to the secant variety to Σ [20]. The pair (Γ, Σ) will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in PG(1, q5), namely those of pseudoregulus and LP type, have been characterized in the literature [6, 30]. Let (Γ, Σ) be a projecting configuration for a maximum scattered linear set in PG(1, q5). Let σ be a generator of G = PΓL(5, q5)Σ, and A = Γ ∩ Γσ4 , B = Γ ∩ Γσ3 . If A and B are not both points, then the projected linear set is of pseudoregulus type [6]. Suppose that they are points. The rank of a point X is the vectorial dimension of the span of the orbit of X under the action of G. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points A and B has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that rk A = rk B = 4. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for q ≤ 25 no new maximum scattered linear set exists.