Towards the classification of maximum scattered linear sets of PG(1,q⁵)

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 $Σ$. 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 by B. Csajbók, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. 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. Then, 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.