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.