In this paper we compute the Waring rank of any polynomial of\ud the form F=M1+...+Mr, where the Mi are pairwise\ud coprime monomials, i.e., GCD(Mi, Mj)=1 for i different from j. In\ud particular, we determine the Waring rank of any monomial. As an\ud application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further\ud application we produce a sum of power decomposition for any form\ud which is the sum of pairwise coprime monomials

We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces

Abstract.We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a 'grove', which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict. Mathematics Subject Classification (2000). 14N15, 51N35.

We ask when certain complete intersections of codimension r can lie on a generic hypersurface in P n . We give a complete answer to this question when 2r ≤ n + 2 in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.