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A refinement of Cauchys theorem on the zeros of quaternion polynomial

Nisar Ahmad Rather, Danish Rashid Bhat, Tanveer Bhat

Abstract

In this paper, we shall present an interesting and significant refinement of a classical result of Cauchy about the moduli of the zeros of a quaternionic polynomial. As an application of this result we shall obtain zero-free regions of polynomials having quaternionic coefficients.

A refinement of Cauchys theorem on the zeros of quaternion polynomial

Abstract

In this paper, we shall present an interesting and significant refinement of a classical result of Cauchy about the moduli of the zeros of a quaternionic polynomial. As an application of this result we shall obtain zero-free regions of polynomials having quaternionic coefficients.

Paper Structure

This paper contains 9 sections, 17 theorems, 29 equations.

Table of Contents

  1. INTRODUCTION
  2. Main Results
  3. Lemma
  4. Proof of the main theorems
  5. Declarations
  6. Availability of data and material
  7. Competing interests
  8. Funding
  9. Author's contributions

Key Result

Theorem A

Let $p(z) = \sum\limits_{j=0}^{n}a_{j}z^{j}$ be a polynomial of degree $n$ such that $0 < a_{0} \leq a_{1} \leq \cdots \leq a_{n}$, then all the zeros of $p(z)$ lie in $\left|z\right| \leq 1$.

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Corollary 3
  • ...and 11 more