Roots and Critical Points of Polynomials over Cayley--Dickson Algebras

We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial $f(x)$ are also roots of its companion polynomial $C_f(x)$ (defined to be the norm of $f(x)$). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$, and we also prove that all roots of $f'(x)$ are contained in the snail of $f(x)$, as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.