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Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation

Prashant Batra

Abstract

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum root-modulus we establish a simple, nearly optimal result.

Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation

Abstract

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum root-modulus we establish a simple, nearly optimal result.

Paper Structure

This paper contains 7 sections, 2 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Known results and thresholds
  3. New inclusion circle via Cassini ovals
  4. Relative overestimations of the new root bound
  5. Overestimation of the maximum root-modulus
  6. Overestimation of the Cauchy bound
  7. Outlook

Key Result

Proposition 2.1

Given a complex polynomial $p$ of degree $n \geq 3$ with Taylor expansion $p(z)= \sum_{i=0}^n a_i z^i$, normalized to be monic $(a_n=1)$ and with largest root-modulus $\mu(p).$ With we have that $\mu(p) \leq \Gamma(p) ,$ where $\Gamma(p)$ is defined as

Theorems & Definitions (2)

  • Proposition 2.1
  • Theorem 3.1