  We provide two new characterizations of bounded orthogonally additive
polynomials from a uniformly complete vector lattice into a convex bornological
space using separately two polynomial identities of Kusraeva involving the root
mean power and the geometric mean. Furthermore, it is shown that a polynomial
on a vector lattice is orthogonally additive whenever it is orthogonally
additive on the positive cone. These results improve recent characterizations
of bounded orthogonally additive polynomials by G. Buskes and the author.
