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Non-archimedean Sendov's Conjecture

Daebeom Choi, Seewoo Lee

TL;DR

Non-archimedean analogue of Sendov’s conjecure is proved and complete list of polynomials over an algebraically closed non-archIMedean field $$K$$ that satisfy the optimal bound in the Sendov's conjecture is provided.

Abstract

We prove non-archimedean analogue of Sendov's conjecure. We also provide complete list of polynomials over an algebraically closed non-archimedean field $K$ that satisfy the optimal bound in the Sendov's conjecture.

Non-archimedean Sendov's Conjecture

TL;DR

Non-archimedean analogue of Sendov’s conjecure is proved and complete list of polynomials over an algebraically closed non-archIMedean field that satisfy the optimal bound in the Sendov's conjecture is provided.

Abstract

We prove non-archimedean analogue of Sendov's conjecure. We also provide complete list of polynomials over an algebraically closed non-archimedean field that satisfy the optimal bound in the Sendov's conjecture.

Paper Structure

This paper contains 3 sections, 5 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The Non-Archimedean Sendov's Conjecture.
  3. Optimality criteria

Key Result

Theorem 1

Let let $r = r_n = |n|^{-1/(n-1)}$. Then the following variation of Sendov's conjecture holds for all $f(z) \in K[z]$: assume that all zeros $z_{i}$ of $f(z)$ are in the closed unit disk centered at zero. Then for each $z_i$, the closed disk $\overline{\mathop{\mathrm{\mathbb{D}}}\nolimits}(z_i, r_n

Theorems & Definitions (15)

  • Conjecture 1: Sendov
  • Theorem 1
  • proof
  • Remark
  • Proposition 1
  • proof
  • Theorem 2
  • Proposition 2
  • proof : First proof of Theorem \ref{['opt_poly_1']}
  • proof : Second proof of Theorem \ref{['opt_poly_1']}
  • ...and 5 more