Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

Given a complex polynomial p(z) with at least three distinct roots, we first prove no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, Bm(z), m ≥ 2. Let θ be a simple root of p(z), V (θ) its Voronoi cell, and Am(θ) its basin of attraction with respect to Bm(z). We prove that given any closed subset C of V (θ), including any homothetic copy of V (θ), there exists m0 such that for all m ≥ m0, C is also a subset of Am(θ). This implies when all roots of p(z) are simple, the basins of attraction of Bm(z) uniformly approximates the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of Bm(z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of Voronoi diagram. In a sense this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.