On the classification of exceptional scattered polynomials

Abstract Let f ( X ) ∈ F q r [ X ] be a q-polynomial. If the F q -subspace U = { ( x q t , f ( x ) ) | x ∈ F q n } 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 , q m r ) 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.