A conjecture that the roots of a univariate polynomial lie in a union of annuli

Details A conjecture that the roots of a univariate polynomial lie in a union of annuli A conjecture that the roots of a univariate polynomial lie in a union of annuli

2006-06-08

arxiv.org/abs/math/0606194v3

Mathematics

Complex Variables

Conjecture 1 in the original version has been resolved. This interim version adds a note to that effect

Scientific paper

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.

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