On the classification of exceptional scattered polynomials

Let f(X) ∈ F_q^r [X] be a q-polynomial. If the Fq-subspace U = {(x^qt, f(x)) | x ∈ F_qⁿ } 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.