Vector lattice powers: continuous and measurable vector functions

In the study of order properties of homogeneous polynomials in vector lattices two constructions are of fundamental importance: the symmetric positive tensor product and the vector lattice power. Both associate a canonical $n$-homogeneous polynomial with each Archimedean vector lattice, such that any other homogeneous polynomial of an appropriate class defined on the same vector lattice is the composition of the canonical polynomial with a linear operator. With this so called “linearization” in hand, various tools of the theory of positive linear operators can be used to study homogeneous polynomials. Thus, the problem of description of the Fremlin symmetric tensor products and the vector lattice powers for special vector lattices arises. The former enables one to study a large class of order bounded homogeneous polynomials, but has a very complicated structure; the latter has a much more transparent structure, but handles a narrower class of homogeneous polynomials, namely orthogonally additive ones. The purpose of this note is to describe the power of the vector lattice of continuous or Bochner measurable vector functions with values in a Banach lattice and to apply this result to the representation of homogeneous orthogonally additive polynomials.