Estimates for polynomial roots

Given a complex polynomial, we obtain lower bounds for the moduli of the roots outside the unit circle. Our main tool is the method of Dandelin-Graeffe, which can be used directly for polynomials with distinct absolute values of the roots. In the general case the arguments of the powers of the roots must be controlled, and we achieve this by two methods: a theorem of Dirichlet, and an argument using linear recurrent sequences.