Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

TL;DR

This paper derives asymptotic formulas and exact counts for self-reciprocal irreducible monic polynomials over finite fields with specific leading coefficients, showing their existence under certain conditions.

Contribution

It provides the first asymptotic expression with explicit error bounds and exact counts for small fields, advancing understanding of polynomial enumeration with constraints.

Findings

Asymptotic expression with explicit error bounds derived

Existence of such polynomials for degree 2n when prescribed coefficients are less than n/4

Exact counts obtained for fields with two or three elements with up to two prescribed coefficients

Abstract

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which is used to show that such polynomials with degree $2n$ always exist provided that the number of prescribed leading coefficients is slightly less than $n/4$. Exact expressions are also obtained for fields with two or three elements and up to two prescribed leading coefficients.