Order properties of homogeneous orthogonally additive polynomials

This is a survey of author's results on the structure of orthogonally additive homogeneous polynomials in vector, Banach and quasi-Banach lattices. The research method is based on the linearization by means of the power of a vector lattice and the canonical polynomial, presented in Section 1. Next, in Section 2, some immediate applications are given: criterion for kernel representability, existence of a simultaneous extension and multiplicative representation from a majorizing sublattice, a characterization of extreme extensions. Section 3 provides a complete description and multiplicative representation for homogeneous disjointness preserving polynomials. Section 4 is devoted to the problem of compact and weakly compact domination for homogeneous polynomials in Banach lattices. Section 5 deals with convexity and concavity of homogeneous polynomials between quasi-Banach lattices, while Section 6 handle the condition under which the quasi-Banach lattice of orthogonally additive homogeneous polynomials is $(p,q)$-convex, or $(p,q)$-concave, or geometrically convex. Section 7 provides a characterization and analytic description of polynomials representable as a finite sum of disjointness preserving polynomials. Finally, some challenging open problems are listed in Section 8.