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

DOI10.1016/J.FFA.2022.102083zbMATH Open1508.11111arXiv2109.09006MaRDI QIDQ6377977

Zhi-Cheng Gao

Publication date: 18 September 2021

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.

Mathematics Subject Classification ID

Polynomials over finite fields (11T06) Exponential sums (11T23) Arithmetic theory of polynomial rings over finite fields (11T55)

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