Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field

TL;DR

This paper improves bounds on counting irreducible and self-reciprocal irreducible polynomials with specific coefficients over finite fields, showing existence conditions for certain degrees.

Contribution

It provides new, tighter error bounds for the enumeration of these polynomials, enhancing understanding of their distribution with prescribed coefficients.

Findings

Improved error bounds for polynomial counts

Existence of self-reciprocal irreducible polynomials under new conditions

Enhanced understanding of polynomial distribution over finite fields

Abstract

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field. The improved bounds imply that self-reciprocal irreducible monic polynomials with degree $2d$ always exist provided that the number of prescribed leading coefficients is slightly less than $ d/2$.