The number of unimodular roots of some reciprocal polynomials

TL;DR

This paper studies the distribution of roots of a special class of reciprocal polynomials, establishing a limit ratio of nonunimodular roots, and extends the analysis to multivariate cases with potential links to Mahler measure.

Contribution

It introduces a sequence of reciprocal polynomials with fixed central coefficients, proves the existence of a limit ratio of nonunimodular roots, and generalizes the results to multivariate polynomials.

Findings

The ratio of nonunimodular roots to degree approaches a limit as degree increases.

An algorithm and numerical method are provided to compute this limit.

Examples suggest a connection to Mahler measure in multivariate polynomials.

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.