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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 scatteredpolynomial of index $t$. The asymptotic behaviour of scattered polynomials ofindex $t$ is an interesting open problem. In this sense, exceptional scatteredpolynomials of index $t$ are those for which $U$ is a maximum scattered linearset in ${\rm PG}(1,q^{mr})$ for infinitely many $m$. The completeclassifications of exceptional scattered monic polynomials of index $0$ (for$q>5$) and of index 1 were obtained by Bartoli and Zhou. In this paper wecomplete the classifications of exceptional scattered monic polynomials ofindex $0$ for $q \leq 4$. Also, some partial classifications are obtained forarbitrary $t$. As a consequence, the complete classification of exceptionalscattered monic polynomials of index $2$ is given.

Towards the full classification of exceptional scattered polynomials