New Estimates and Existence Results About Irreducible Polynomials and Self-Reciprocal Irreducible Polynomials with Prescribed Coefficients Over a Finite Field

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 numbers of irreducible monic polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field $${\mathbb F}_{q}$$ . The new lower bounds are used to derive some existence results about irreducible monic polynomials of degree d and self-reciprocal irreducible monic polynomials of degree 2d with roughly d/2 coefficients prescribed at positions including the middle range $$d/2-\log _q d\le j\le d/2+\log _q d$$ .