Nilpotent linearized polynomials over finite fields, revisited

In this paper we develop further studies on nilpotent linearized polynomials (NLP's) over finite fields, a class of polynomials introduced by the second author. We characterize certain NLP's that are binomials and show that, in general, NLP's are also nilpotent over a particular tower of finite fields. We also develop results on the construction of permutation polynomials from NLP's, extending some past results. In particular, the latter yields polynomials that permutes certain infinite subfields of ${\bar{F}}_q$ and have a very particular cycle structure. Finally, we provide a nice correspondence between certain NLP's and involutions in binary fields and, in particular, we discuss a general method to produce affine involutions over binary fields without fixed points.