Characterization and multiplicative representation of homogeneous disjointness preserving polynomials

Let $E$ and $F$ be vector lattices and $P\colon E\to F$ an order bounded orthogonally additive (i.e. $|x|\wedge|y|=0$ implies $P(x+y)=P(x)+P(y)$ for all $x,y\in E$) $s$-homogeneous polynomial. $P$ is said to be disjointness preserving if its corresponding symmetric $s$-linear operator from $E^s$ to $F$ is disjointness preserving in each variable. The main results of the paper read as follows:
Theorem 3.9. The following are equivalent: (1) $P$ is disjointness preserving; (2) $\hat d^nP(x)(y)=0$ and $Px\perp Py$ for all $x,y\in E$, $x\perp y$, and $1\leq n<s$; (3) $P$ is orthogonally additive and $x\perp y$ implies $Px\perp Py$ for all $x,y\in E$; (4) {\it there exist a vector lattice $G$ and lattice homomorphisms $S_1,S_2\colon E \to G$ such that $G^{s\scriptscriptstyle\odot}\subset F$, $S_1(E)\perp S_2(E)$, and $Px=(S_1x)^{s\scriptscriptstyle\odot}-(S_2x)^{s\scriptscriptstyle\odot}$ for all $x\in E$}; (5) {\it there exists an order bounded disjointness preserving linear operator $T:E^{s\scriptscriptstyle\odot}\to F$ such that $Px=T(x^{s\scriptscriptstyle\odot})$ for all $x\in E$}.
Theorem 4.7. {\it Let $E$ and $F$ be Dedekind complete vector lattices. There exists a partition of unity $(\rho_\xi)_{\xi\in\Xi}$ in the Boolean algebra of band projections $\mathfrak P(F)$ and a family $(e_\xi)_{\xi\in\Xi}$ in $E_+$ such that $P(x)=o$-$\sum_{\xi\in\Xi}W\circ\rho_\xi S(x/e_\xi)^{s\scriptscriptstyle\odot}$ $(x\in E)$, where $S$ is the shift of $P$ and $W\colon\mathscr F\to\mathscr F$ is the orthomorphism multiplication by $o$-$\sum_{\xi\in\Xi}\rho_\xi P(e_\xi)$.