un 2 00 6 A conjecture that the roots of a univariate polynomial lie in a union of annuli ∗

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. 1 Conjecture concerning annuli Let p(z) be a univariate polynomial with coefficients in C. Let z1, . . . , zn be its roots. Let ζ1, . . . , ζn−1 be the roots p . This paper proposes the conjecture that z1, . . . , zn lie in a union of n − 1 annuli, one for each of ζ1, . . . , ζn−1. The two radii of each annulus are determined from higher derivative values, and the inner radius is a constant fraction of the outer radius. The formal statement is as follows. Supported in part by NSF award 0434338. Department of Computer Science, 4130 Upson Hall, Cornell University, Ithaca, NY 14853 USA, [email protected].