Exceptional scattered polynomials

Let f be an Fq -linear function over Fqn. If the Fq-subspace U = {(xqt , f (x)) : x ∈ Fqn } 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 PG(1, qmn ) for infinitely many m. Our main results are the 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.