In this paper, we study polynomial norms, i.e. norms that are the dth root of a degree-d homogeneous polynomial f . We first show that a necessary and sufficient condition for f to be a norm is for f to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from dth roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polyn...