Singularities of linear systems and the Waring Problem

Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper I answer this question when the degree of $f$ is greater than the number of variables. To do this I translate the algebraic statement into a geometric one concerning the singularities of linear systems of $P^n$ with assigned singularities.