  In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the
multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$
such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a
consequence we solve the Waring Problem for all monomials, i.e. we compute the
minimal number of linear forms needed to write a monomial as a sum of powers of
these linear forms. Moreover, we give an explicit description of a sum of
powers decomposition for monomials. We also produce new bounds for the Waring
rank of polynomials which are a sum of pairwise coprime monomials.
