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On the Enestrom-Kakeya Theorem for polynomials of an octonionic variable

Ting Yang, Xinyuan Dou

Abstract

To study the zeros of octonionic polynomials, we generalize the well-known Enestrom-Kakeya Theorem to the case of octonions. In this paper, we first deal with octonionic polynomials with nonnegative and monotonic coefficients, and prove that its zero set is contained within the closed sphere of octonion space. Then, we also consider the octonionic polynomials which the coefficients muduli is monotonic and the real parts of the coefficients is monotonic respectively, and get some results.

On the Enestrom-Kakeya Theorem for polynomials of an octonionic variable

Abstract

To study the zeros of octonionic polynomials, we generalize the well-known Enestrom-Kakeya Theorem to the case of octonions. In this paper, we first deal with octonionic polynomials with nonnegative and monotonic coefficients, and prove that its zero set is contained within the closed sphere of octonion space. Then, we also consider the octonionic polynomials which the coefficients muduli is monotonic and the real parts of the coefficients is monotonic respectively, and get some results.
Paper Structure (3 sections, 10 theorems, 63 equations)

This paper contains 3 sections, 10 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Priliminaries
  3. Main results

Key Result

Theorem 1.1

Let $p(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ with real coefficients satisfying then all the zeros of $p$ lie in $|z|\leq 1.$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 10 more