Homogeneous polynomials, root mean power, and geometric means in vector lattices

It is proved that for a homogeneous orthogonally additive polynomial $P$ of degree $s\in\mathbb N$ from a uniformly complete vector lattice $E$ to some convex bornological space the equations $P(\mathfrak S_s(x_1,\ldots,x_N))= P(x_1)+\ldots+P(x_N)$ and $P(\mathfrak G(x_1,\ldots,x_s))=\check P(x_1,\ldots,x_s)$ hold for all positive $x_1,\ldots,x_s\in E$, where $\check P$ is an $s$-linear operator generating $P$, while $\mathfrak S_s(x_1,\ldots,x_N)$ and $\mathfrak G(x_1,\ldots,x_s)$ stand respectively for root mean power and geometric mean in the sense of homogeneous functional calculus.