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Polynomial root-finding algorithms and branched covers

Myong-Hi Kim, Scott Sutherland

TL;DR

The work tackles efficient numerical root-finding for degree-$d$ polynomials by computing an $\epsilon$-factorization (an approximate factorization into linear factors) through a path-lifting method grounded in the polynomial's branched-cover structure. By combining Smale's path-lifting with a global topological view, the authors enable simultaneous tracking of multiple roots, reducing arithmetic work compared to root-by-root approaches. They present a concrete algorithm that locates $d/2$ roots at a time and prove a worst-case arithmetic complexity bound of $O(d(d)|\epsilon| + d^2(d)^2)$ for achieving an $\epsilon$-factorization, with bit-complexity analysis deferred to a future work. The method is highlighted as stable under rounding errors, amenable to parallelization, and practical for moderate degrees and precision, offering a scalable alternative to existing algorithms in the numerical algebraic setting.

Abstract

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an $ε$-factorization of the polynomial which has an arithmetic complexity of $\Order{d^2(\log d)^2 + d(\log d)^2|\logε|}$. At the present time (1993), this complexity is the best known in terms of the degree.

Polynomial root-finding algorithms and branched covers

TL;DR

The work tackles efficient numerical root-finding for degree- polynomials by computing an -factorization (an approximate factorization into linear factors) through a path-lifting method grounded in the polynomial's branched-cover structure. By combining Smale's path-lifting with a global topological view, the authors enable simultaneous tracking of multiple roots, reducing arithmetic work compared to root-by-root approaches. They present a concrete algorithm that locates roots at a time and prove a worst-case arithmetic complexity bound of for achieving an -factorization, with bit-complexity analysis deferred to a future work. The method is highlighted as stable under rounding errors, amenable to parallelization, and practical for moderate degrees and precision, offering a scalable alternative to existing algorithms in the numerical algebraic setting.

Abstract

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an -factorization of the polynomial which has an arithmetic complexity of . At the present time (1993), this complexity is the best known in terms of the degree.

Paper Structure

This paper contains 5 sections, 4 theorems, 9 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Root and coefficient bounds
  3. Approximate zeros
  4. The path lifting method
  5. Branched covers, inverse functions, and all that

Key Result

lemma 1

All the zeros of the polynomial $z^d + \sum_{j=0}^{d-1} a_j z^j$ lie within the open disk with center 0 and radius

Figures (2)

  • Figure 1: The source and target spaces in the path lifting method. In the source space, each $z_i$ is indicated by a black dot, and $f(z_i)$ is indicated by a black dot in the target space. Similarly, the $w_i$ are indicated by tick marks in the target space, and $f^{-1}(w_i)$ by ticks in the source space.
  • Figure :

Theorems & Definitions (7)

  • lemma 1
  • Definition 1
  • lemma 2
  • lemma 3
  • remark 1
  • proof
  • theorem 2