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Expansivity theory and Sendov's conjecture

Theophilus Agama

Abstract

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring $\mathbb{C}[x]$. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. As an application we give a proof of Sendov's conjecture. We start by establishing the uniformly diminishing state of the mass of an expansion.

Expansivity theory and Sendov's conjecture

Abstract

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring . As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. As an application we give a proof of Sendov's conjecture. We start by establishing the uniformly diminishing state of the mass of an expansion.

Paper Structure

This paper contains 39 sections, 53 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Tuple viewpoint:
  3. Expansion/recovery as a dynamical pair:
  4. Mass, momentum and index:
  5. Geometry via a boundary integral:
  6. Compact and regular phases; consequences for Sendov:
  7. Additional notations
  8. Calculus on tuples on tuples of polynomials
  9. Differentiation on tuples of polynomials
  10. Properties of differentiation on tuples of polynomials
  11. Integration on tuples of polynomials
  12. Expansion on a tuple of polynomials
  13. Phase expansions
  14. The rank of an expansion
  15. The limit of an expansion
  16. ...and 24 more sections

Key Result

Proposition 3.2

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be tuples of the elements in $\mathbb{R}[x]$ and $c\in \mathbb{R}$. We have the following properties:

Theorems & Definitions (147)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Definition 4.1
  • Remark 4.2
  • Proposition 4.3
  • proof
  • ...and 137 more