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Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $Σ$ of $\mathrm{PG}(4,q^5)$ 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 $\mathrm{PG}(1,q^5)$, 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 $\mathrm{PG}(1,q^5)$, let $σ$ be a generator of $\mathbb{G}=\mathrm{P}Γ\mathrm{L}(5,q^5)_Σ$, and $A=Γ\capΓ^{σ^4}$, $B=Γ\capΓ^{σ^3}$. 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 $\mathbb{G}$. 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 $\mathrm{rk} A=\mathrm{rk} B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\leq 25$, no new maximum scattered linear set exists.