Generalizations of self-reciprocal polynomials

Abstract

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.

On the existence and number of invariant polynomials

“…. The number of monic irreducible polynomials satisfying the previous identity was obtained in Corollary 7 of [3]. Combining this corollary with Theorem 2.7, we easily obtain the following result.…”

Möbius–frobenius Maps on Irreducible Polynomials

“…From Theorem 4.1, both g and σ 1 (g) are SCRIM polynomials of degree n. The construction of self-reciprocal polynomials over finite fields is discussed in [5] In particular, if [B] ∈ PGL(2, q) is an involution and n > 1 is odd, the number of n-degree [B, σ 1 ]-invariants over F q 2 is exactly twice the number of 2n-degree [B]-invariants over F q . Exact enumeration formulas for the number of [B]-invariants for a generic involution [B] ∈ PGL(2, q) are given in [4].…”

“…Many authors have studied this and other similar actions, focusing on the characterisation and number of [A]-invariants, that is, monic irreducible polynomials f for which [A] • f = f (see [3,4,6,7]).…”

A group action on multivariate polynomials over finite fields

“…We divide the proof of this result in two main cases. We first consider [A] an element of type t ≤ 3: these cases can be easily covered by the works in [2] and [4]. For the case t = 4, we generalize the main technique employed in [2].…”

“…We first consider [A] an element of type t ≤ 3: these cases can be easily covered by the works in [2] and [4]. For the case t = 4, we generalize the main technique employed in [2]. In every case, we first consider an element of type t in reduced form and then we apply Lemma 2.5.…”