On the rank $1$ decompositions of symmetric tensors

Abstract

Here we study the uniqueness of a representation of a homogeneous polynomialas a sum of a small number of powers of linear forms (equivalently, a representation of a symmetric tensoras a sum of powers) or (when it is not unique) describe all such additive decompositions. We requirea linear upper bound for the number of addenda with respect to the degree of the polynomial and, for someresults, assumptions like linearly general position.