Generalizations of self-reciprocal polynomials

TL;DR

This paper explains and simplifies the extension of Carlitz's formula for counting irreducible self-reciprocal polynomials over finite fields, including new results on polynomials generated by quadratic and higher-degree transformations.

Contribution

It provides a unified explanation and a simpler proof for the extension of Carlitz's formula, along with new results on polynomials from quadratic and higher-degree transformations.

Findings

Simplified proof of Ahmadi's extension of Carlitz's formula

New results on polynomials from quadratic transformations

Extension to transformations of higher degree

Abstract

A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.