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Limited polynomials and sendov's conjecture

Theophilus Agama

Abstract

In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign.

Limited polynomials and sendov's conjecture

Abstract

In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign.

Paper Structure

This paper contains 6 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction and motivation
  2. Problem statement
  3. Limited complex-valued polynomials
  4. Properties of limited polynomials
  5. Critical points of limited polynomials
  6. Further remarks

Key Result

Proposition 3.2

Let $P(z)$ and $Q(z)$ be any $\epsilon$ and $\delta$ limited polynomials, respectively. If $\mathcal{Z}(P(z))\cap \mathcal{Z}(Q(z))=\emptyset$, where $\mathcal{Z}(P(z))$ and $\mathcal{Z}(Q(z))$ are the set of zeros of $P(z)$ and $Q(z)$, respectively, then the product $P(z)Q(z)$ is $\epsilon \delta$-

Theorems & Definitions (25)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 4.1
  • proof
  • ...and 15 more