Roots and critical points of polynomials over Cayley–Dickson algebras

The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial (Formula presented.). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of (Formula presented.) belong to the convex hull of the roots of (Formula presented.), and we also show that all roots of (Formula presented.) are contained in the snail of f(x), as defined by Ghiloni and Perotti.