On the zeros of polynomials with complex coefficients

Introduction. Problems in dynamics very frequently have physically realizable solutions only if the determinantal equation of the system has all its roots in the negative half of the complex plane. It is therefore convenient to have a simple algorithm for testing whether this condition holds without actually computing the roots. Solutions to this problem have been considered by Cauchy [l], 2 Routh [], and many others. Hurwitz [4] gave a method for polynomials with real coefficients of the form (1.1) P(z) z n + aiz n ~l + azz"-* H + a n According to his rule, all of the roots lie in the half-plane R(z) <0 if and only if all the determinants D* = 01,