Investigating the exceptionality of scattered polynomials

Abstract

Scattered polynomials over a finite field Fqn have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-qt-partially scattered and R-qt-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-qt-partially scattered and R-qt-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-qt-partially scattered one. We investigate a large family F of R-qt-partially scattered polynomials, containing examples of exceptional R-qt-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-qt-partially scattered. Many polynomials in F are inequivalent and geometric arguments are used to determine their equivalence classes under the action of ΓL(2n/t,qt).