Kuhn’s Method for Finding All Roots of Univariate Polynomials

Abstract

In this chapter we consider the problem of finding all roots of a monic polynomial f (z)of degree n in the complex variable z with complex numbers as coefficients. As it is well-known, the fundamental theorem of algebra asserts that the polynomial f (z)has at least one root. See for example van der Waerden [1970]. There are many non-constructive proofs for this theorem in the literature. The problem of finding all roots of a polynomial is not only interesting in its theoretical nature but also very important in many engineering designs such as, to mention a few, control system design and telecommunication system design. Kuhn [1974, 1977] developed a remarkable algorithm which is guaranteed to find all roots of a polynomial, whereas all the existing methods before Kuhn’s method are not guaranteed to find all roots. Kuhn’s method sets yet another powerful example of simplicial algorithms. Briefly speaking, Kuhn’s method is based on both a continuously refining triangulation of the cross product of the complex plane ℂ and an interval [−1, +∞), and a special integer labeling rule. The algorithm starts with n entrances at level ℂ × {−1}, and generates n paths of simplices which lead to n approximate roots of any a priori chosen accuracy within a finite number of steps. By taking limit, the n paths lead to n roots of the polynomial f (z).Kuhn’s method will be used as a subroutine for finding all solutions of multivariate polynomial equations to be discussed in the next chapter.