## A representation theorem for orthogonally additive polynomials on Riesz spaces.(English)Zbl 1297.46006

The purpose of this paper is to prove a representation theorem for orthogonally additive polynomials acting on Riesz spaces. The idea uses the notion of $$p$$-orthosymmetric multilinear forms which is introduced. Moreover, the authors show that the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values in a uniformly complete Archimedean Riesz space is isomorphic to the space of positive linear forms on the $$n$$-power in the sense of [K. Boulabiar and G. Buskes, Commun. Algebra 34, No. 4, 1435–1442 (2006; Zbl 1100.46001)] of the original Riesz space.
It should be noted that the reviewer has obtained an alternative representation theorem for homogeneous orthogonally additive polynomials on Riesz spaces. Its relation with the results presented here is discussed in [M. A. Toumi, Bull. Belg. Math. Soc. - Simon Stevin 20, No. 4, 621–638 (2013; Zbl 1280.06014)].

### MSC:

 46A40 Ordered topological linear spaces, vector lattices 46G25 (Spaces of) multilinear mappings, polynomials 47B65 Positive linear operators and order-bounded operators