A Classification of Permutation Polynomials through Some Linear Maps

Abstract

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the generalized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes of permutation polynomials studied in literature neatly fall into classes defined using these linear maps. We characterize the shapes of permutation polynomials that appear in the various generalized eigenspaces of these linear maps. For the case of