Towards the full 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 behaviour 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 mr ) for infinitely many m. The complete 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 complete classification of exceptional scattered monic polynomials of index 2 is given.

A new family of maximum scattered linear sets in $\mathrm{PG}(1,q^6)$

“…We point out his construction in the last section. By the results of [1] and [2], it seems that examples of maximum scattered linear sets are rare.…”