The quaternionic Gauss-Lucas Theorem

Abstract

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge 2$ 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 $d\ge 2$, together with some consequences.

7 pages, 1 figure. Remarks added in section 3. Proposition 14 added with complete proof

References in corpus (1)

• On the Gauss-Lucas theorem in the quaternionic setting

Cited by in corpus (3)

• Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
• The Bernstein inequality for slice regular polynomials
• A four dimensional Bernstein Theorem