Sums of powers of orthogonally additive polynomials

Abstract

It is proved that for a given two positive integers N,n with N<n and N orthoregular homogeneous polynomials of the same degree acting between Archimedean vector lattices and pairwise independent in some sense, the sum of these polynomials each raised to the power of n and multiplied by an orthomorphism, is orthogonally additive if and only if all these polynomials multiplied by the corresponding orthomorphisms are disjointness preserving. It is also shown that given three positive integers N,n,m, an order bounded orthogonally additive polynomial with values in a Dedekind complete vector lattice is the sum of n-th powers of N order bounded disjointness preserving m-homogeneous polynomials if and only if it is representable as a disjoint sum of N disjointness preserving mn-homogeneous polynomials.