  We conjecture that the roots of a degree-n univariate complex polynomial are
located in a union of n-1 annuli, each of which is centered at a root of the
derivative and whose radii depend on higher derivatives. We prove the
conjecture for the cases of degrees 2 and 3, and we report on tests with
randomly generated polynomials of higher degree.
  We state two other closely related conjectures concerning Newton's method. If
true, these conjectures imply the existence of a simple, rapidly convergent
algorithm for finding all roots of a polynomial.
