A presentation of a degree d form in n+1 variables as the sum of
homogenous elements ``essentially'' involving n variables is called a {\em
codimension one decomposition}. Codimension one decompositions are introduced
and the related Waring Problem is stated and solved. Natural schemes describing
the codimension one decompositions of a generic form are defined. Dimension and
degree formulae for these schemes are derived when the number of summands is
the minimal one; in the zero dimensional case the scheme is showed to be
reduced. These results are obtained by studying the Chow variety Ξn,sβ
of zero dimensional degree s cycles in \PP^n. In particular, an explicit
formula for degΞn,sβ is determined