Non-archimedean Sendov's Conjecture

TL;DR

This paper establishes a non-archimedean version of Sendov's conjecture and classifies polynomials over algebraically closed non-archimedean fields that meet the conjecture's bounds.

Contribution

It introduces a non-archimedean analogue of Sendov's conjecture and provides a complete classification of polynomials satisfying the optimal bounds.

Findings

Proved the non-archimedean analogue of Sendov's conjecture.

Provided a complete list of polynomials meeting the bounds.

Extended classical conjecture to non-archimedean fields.

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.