Exceptional Scattered Polynomials

Abstract: Let $f$ be an $\mathbb{F}q$-linear function over $\mathbb{F}{q^n}$. If the $\mathbb{F}q$-subspace $U= { (x^{q^t}, f(x)) : x\in \mathbb{F}{q^n} }$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.