The additive index of polynomials over finite fields

Abstract

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We show that every polynomial P(x)∈Fq[x] can be expressed uniquely in its additive index form such that P(x)=f(L(x))+M(x) where L(x),M(x) are p-linearized polynomials over Fq, deg⁡(M)<deg⁡(L), L(x) splits completely over Fq and L is of the maximal degree. As applications, we study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterization of permutation polynomials.