Self-reciprocal irreducible polynomials with prescribed coefficients
Distribution of irreducible polynomials with additional properties over finite fields has attracted a lot of attention. This paper provides estimates for the number of self-reciprocal monic irreducible polynomials that having the \(t\) lower degree coefficients fixed to given values, over finite fields of odd characteristic. In particular, the existence of a self-reciprocal monic irreducible polynomial of degree \(2m\) with prescribed \(t\) low degree coefficients for \(t \leq m/2 - \log_q(2m) -1\) is established. The approach used in this paper is based on the work of \textit{L. Carlitz} [J. Reine Angew. Math. 227, 212--220 (1967; Zbl 0155.09801)], who has described monic self-reciprocal irreducible polynomials in terms of the form \(x^m P(x+x^{-1})\), where \(P\) is a monic irreducible polynomial of degree \(m\) such that \(x^2 -4\) is a non-square modulo \(P\), and also has enumerated of these polynomials using Dirichlet \(L\)-functions.
scientific article; zbMATH DE number 5868431
scientific article