On the Theorem of Gauss--Lucas for quaternions

TL;DR

This paper extends the Gauss-Lucas theorem to quaternionic polynomials, showing that the roots of the derivative are contained within specific projected regions, thus strengthening previous quaternionic results.

Contribution

It provides a new proof that the roots of quaternionic polynomial derivatives lie within certain projected sets, enhancing the quaternionic Gauss-Lucas theorem.

Findings

Roots of quaternionic polynomial derivatives are contained in specific projected sets

Strengthens the quaternionic Gauss-Lucas theorem from 2018

Provides a new geometric interpretation of roots in quaternionic polynomials

Abstract

It is proved that the roots of the derivative of a polynomial with quaternionic coefficients belong to the union of the intersections of sets defined in terms of certain projections of a polynomial. The result strengthens the quaternion version of Gauss-Lucas theorem, proved by Ghiloni and Perotti in 2018.