Algebraic Characterization of Polynomials Whose Zeros Lie in Certain Algebraic Domains

A new algebraic criterion is given for a polynomial phi with complex coefficients to have all its zeros in a certain type of algebraic region T of the complex plane. In particular, T may be any circle or half plane. The criterion is effectively computable from the coefficients of the polynomial phi. The classical results of Hermite, Hurwitz, Lyapunov, Schur-Cohn, and others appear as special cases of the new criterion.