Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuous approximate roots of polynomial equations via shape theory

Joshua Lau, Vicente Marin-Marquez

TL;DR

This work develops a topological framework for solving monic polynomials over $C(X)$ by translating root-finding into lifting problems in covering spaces and then into pro-group morphisms into braid groups. Using shape theory, the authors define the fundamental pro-group $_1(X)$ and prove that for $ ext{dim}(X) obreak o obreak 1$, $C(X)$ is approximately algebraically closed precisely when every induced map $_1(X) o  B_n$ lands in the pure braid subgroup, for all $n$. They further show that polynomials with no repeated roots provide a stable setting where approximate roots converge to exact roots under mild hypotheses, and they derive perturbation results guaranteeing density of polynomials with no repeated roots in coefficient space. The paper also links these topological criteria to co-existentially closed continua and provides explicit low-degree (quadratic, cubic, quartic) results and counterexamples, illustrating when homotopical invariants suffice to solve polynomials and when obstructions persist. Overall, the work bridges shape theory, braid-group topology, and continuous algebraic closure, offering concrete criteria and examples for when continuous root finding is possible in one-dimensional continua and highlighting the limitations at degree four and higher.

Abstract

We study continuous approximate solutions to polynomial equations over the ring $C(X)$ of continuous complex-valued functions over a compact Hausdorff space $X$. We show that when $X$ is one-dimensional, the existence of such approximate solutions is governed by the behaviour of maps from the fundamental pro-group of $X$ into braid groups.

Continuous approximate roots of polynomial equations via shape theory

TL;DR

This work develops a topological framework for solving monic polynomials over by translating root-finding into lifting problems in covering spaces and then into pro-group morphisms into braid groups. Using shape theory, the authors define the fundamental pro-group and prove that for , is approximately algebraically closed precisely when every induced map lands in the pure braid subgroup, for all . They further show that polynomials with no repeated roots provide a stable setting where approximate roots converge to exact roots under mild hypotheses, and they derive perturbation results guaranteeing density of polynomials with no repeated roots in coefficient space. The paper also links these topological criteria to co-existentially closed continua and provides explicit low-degree (quadratic, cubic, quartic) results and counterexamples, illustrating when homotopical invariants suffice to solve polynomials and when obstructions persist. Overall, the work bridges shape theory, braid-group topology, and continuous algebraic closure, offering concrete criteria and examples for when continuous root finding is possible in one-dimensional continua and highlighting the limitations at degree four and higher.

Abstract

We study continuous approximate solutions to polynomial equations over the ring of continuous complex-valued functions over a compact Hausdorff space . We show that when is one-dimensional, the existence of such approximate solutions is governed by the behaviour of maps from the fundamental pro-group of into braid groups.

Paper Structure

This paper contains 12 sections, 34 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spaces of polynomials and their solutions
  3. Spaces of polynomials
  4. Continuous roots of polynomials
  5. Stability and perturbation
  6. Shape theory
  7. Pro-categories
  8. Shape theory
  9. Eilenberg-MacLane spaces
  10. Polynomials via $\ul{\pi}_1(X)$
  11. Main results and some examples
  12. Low-degree polynomials and braid groups

Key Result

Theorem 1

Suppose that $X$ is a continuum with covering dimension at most one. Then $C(X)$ is approximately algebraically closed if and only if for every $n \geq 1$, the image of any morphism $\ul{\pi}_1(X) \to \mathcal{B}_n$ lies in the subgroup of pure braids, i.e. in the kernel of the canonical map $\mathc

Figures (2)

  • Figure 1: Illustrated is the $1$-skeleton for $\widetilde{Y}$ along with labels for its six $0$-cells. The $1$-cells associated to $S$ are drawn in black, while the $1$-cells associated to $Q$ are drawn in blue.
  • Figure 2: On the left we illustrate the loop on $\widetilde{Y}$ corresponding to $U$ that starts at $\widetilde{y_0}$. On the right we depict this loop once we collapse $\widetilde{Y}$ to the theta space. If we use $a$ and $b$ to denote the loops on the theta space by traversing the respective semi-circles counterclockwise, we see that the loop corresponding to $U$ is $ab^{-1}$. A similar calculation shows that $V$ corresponds to $bab^{-1}$, and these two loops are freely generate the fundamental group.

Theorems & Definitions (73)

  • Theorem 1: Theorem \ref{['thm:low_dim_approx_alg_closure_propi1']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 63 more