root-free annulus one uses a consequence of the Rouché theorem: For k = 1, ..., n − 1 the polynomial equation u > 0, has, by Descartes' rule of signs zero or two positive roots . In the latter case, there are exactly k roots inside the (closed) disk and is a root-free (open) annulus. References Schönhage, Arnold (1982): The fundamental theorem of algebra in terms of computational complexity. Preliminary Report, Math. Inst. Univ. Tübingen (1982), 49 pages. (ps.gz) Pan, Victor (1998). Algorithm for Approximating Complex Polynomial Zeros Pan, Victor (2002). Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding Magma documentation. Real and Complex Fields: Element Operations. Root-finding algorithms