  The aim of this article is to prove a representation theorem for orthogonally
additive polynomials in the spirit of the recent theorem on representation of
orthogonally additive polynomials on Banach lattices but for the setting of
Riesz spaces. To this purpose the notion of $p$--orthosymmetric multilinear
form is introduced and it is shown to be equivalent to the orthogonally
additive property of the corresponding polynomial. Then the space of positive
orthogonally additive polynomials on an Archimedean Riesz space taking values
on an uniformly complete Archimedean Riesz space is shown to be isomorphic to
the space of positive linear forms on the $n$-power in the sense of Boulabiar
and Buskes of the original Riesz space.
