The number of unimodular roots of some reciprocal 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.