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 $\Sigma$ of $\mathrm{PG}(4,q^5)$ from a plane $\Gamma$ external to the secant variety to $\Sigma$. The pair $(\Gamma,\Sigma)$ 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. Csajb\'{o}k, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. Let $(\Gamma,\Sigma)$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$, let $\sigma$ be a generator of $\mathbb{G}=\mathrm{P}\Gamma \mathrm{L}(5,q^5)_\Sigma$, and $A=\Gamma\cap\Gamma^{\sigma^4}$, $B=\Gamma\cap\Gamma^{\sigma^3}$. 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.