Roots and critical points of polynomials over Cayley–Dickson algebras

Abstract

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 . 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 belong to the convex hull of the roots of , and we also show that all roots of are contained in the snail of f(x), as defined by Ghiloni and Perotti.