Towards the full classification of exceptional scattered polynomials

Let $f(X) \in \mathbb{F}_{q^r}[X]$ be a $q$-polynomial. If the $\mathbb{F}_q$-subspace $U=\{(x^{q^t},f(x)) \mid x \in \mathbb{F}_{q^n}\}$ defines a maximum scattered linear set, then we call $f(X)$ a scattered polynomial of index $t$. The asymptotic behaviour of scattered polynomials of index $t$ is an interesting open problem. In this sense, exceptional scattered polynomials of index $t$ are those for which $U$ is a maximum scattered linear set in ${\rm PG}(1,q^{mr})$ for infinitely many $m$. The complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index 1 were obtained by Bartoli and Zhou. In this paper we complete the classifications of exceptional scattered monic polynomials of index $0$ for $q \leq 4$. Also, some partial classifications are obtained for arbitrary $t$. As a consequence, the complete classification of exceptional scattered monic polynomials of index $2$ is given.