The solution to the Waring problem for monomials and the sum of coprime monomials

Abstract

In this paper we compute the Waring rank of any polynomial of the form $F={\sum }_{i=1}^r{M}_i$, where the $M_i$ are pairwise coprime monomials, i.e., $\mathit{GCD}(M_i,M_j)=1$ for $i\ne j$. In particular, we determine the Waring rank of any monomial. As an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials.