The quaternionic Gauss-Lucas Theorem

TL;DR

This paper explores the extension of the Gauss-Lucas Theorem to quaternions, revealing limitations of the classic reformulation and proposing a new version applicable for all degrees, with implications for quaternionic polynomial analysis.

Contribution

It introduces a novel quaternionic Gauss-Lucas Theorem valid for all degrees, overcoming limitations of previous reformulations that only held for quadratic polynomials.

Findings

The classic quaternionic reformulation holds only for degree 2.

A new quaternionic Gauss-Lucas Theorem is established for all degrees.

Consequences of the new theorem are discussed.

Abstract

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge2$ has a natural reformulation over quaternions, obtained via rotation around the real axis. We prove that such a reformulation is true only for $d=2$. We present a new quaternionic version of the Gauss-Lucas Theorem valid for all $d\geq2$, together with some consequences.