On the classification of exceptional scattered polynomials

Let f(X)∈Fq[X] be a q-polynomial. If the Fq-subspace U={(xqjavax.xml.bind.JAXBElement@7c6eb7af,f(x))|x∈Fqjavax.xml.bind.JAXBElement@f292279} defines a maximum scattered linear set, then we call f(X) a scattered polynomial of index t. The asymptotic behavior 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 PG(1,qmr) for infinitely many m. The classifications of exceptional scattered monic polynomials of index 0 (for q>5) and of index 1 were obtained in [1]. In this paper we complete the classifications of exceptional scattered monic polynomials of index 0 for q≤4. Also, some partial classifications are obtained for arbitrary t. As a consequence, the classification of exceptional scattered monic polynomials of index 2 is given.