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DOI: 10.23671/VNC.2014.4.10260

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

Kusraeva, Z. A.
Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 4.
Abstract:
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.
Keywords: vector lattice, homogeneous polynomial, linearization of a polynomial, root mean power, geometric mean