2019/05/27 by Bartoli, Daniele, Montanucci, Maria
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.11390
Let f(X) ∈ \mathbbFqr[X] be a q-polynomial. If the \mathbbFq-subspace U=\(xqt,f(x)) | x ∈ \mathbbFqn\ 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 \rm PG(1,qmr) for infinitely many m. The complete classifications of exceptional scattered monic polynomials of index 0 (for q>5) and of index 1 were obtained by Bartoli and Zhou. 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.