Prescribing coefficients of invariant irreducible polynomials

Let F-q be the finite field of q elements. We define an action of PGL(2, q) on F-q[X] and study the distribution of the irreducible polynomials that remain invariant under this action for lower-triangular matrices. As a result, we describe the possible values of the coefficients of such polynomials and prove that, with a small finite number of possible exceptions, there exist polynomials of given degree with prescribed high degree coefficients. (C) 2017 Elsevier Inc. All rights reserved.