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=∑i=1rMi, where the Mi are pairwise coprime monomials, i.e., GCD(Mi,Mj)=1 for i≠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.