On the expansivity gap of integer polynomials

Abstract

Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these polynomials, assuming that the coefficients are integers. We give lower bounds on the expansivity gap, using the degree and the coefficient size as parameters. We also construct a family of polynomials which indicate the sharpness of these bounds. As a side-result, we present an explicit condition for deciding expansivity of polynomials, which we find superior to the existing recursive methods for our purpose.