The number of unimodular roots of some reciprocal polynomials

2019/09/27 by Stankov, Dragan
#11R06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1909.12986

Abstract

We introduce a sequence P_2n of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of P_2n and its degree d has a limit when d tends to infinity. We present an algorithm for calculation the limit and a numerical method for its approximation. If P_2n is the sum of a fixed number of monomials we determine the central coefficients such that the ratio has the minimal limit. We generalise the limit of the ratio for multivariate polynomials. Some examples suggest a theorem for polynomials in two variables which is analogous to Boyd's limit formula for Mahler measure.

Related

• The number of nonunimodular roots of a reciprocal polynomial
• Approximation of the number of nonunimodular zeros of a self-reciprocal polynomial
• Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field
• Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field
• The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set