On the Gauss-Lucas theorem in the quaternionic setting

TL;DR

This paper provides a complete proof of the quaternionic analogue of the Gauss-Lucas theorem, showing that polynomial critical points relate to the convex hull of zeros of the symmetrized polynomial.

Contribution

It offers a new, complete proof of the quaternionic Gauss-Lucas theorem and discusses its implications.

Findings

Critical points lie in the convex hull of zeros of the symmetrized polynomial

Complete proof clarifies the quaternionic Gauss-Lucas relation

Discussion of consequences for quaternionic polynomial theory

Abstract

In theory of one complex variable, Gauss-Lucas Theorem states that the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the polynomial. The exact analogue of this result cannot hold, in general, in the quaternionic case; instead, the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the so-called symmetrization of the given polynomial. An incomplete proof of this statement was given in [8]. In this paper we present a different but complete proof of this theorem and we discuss a consequence.