Waring loci and decompositions of low rank symmetric tensors

Given a symmetric tensor, i.e., a homogeneous polynomial, a Waring decomposition is an expression as sum of symmetric decomposable tensors, i.e., powers of linear forms. We call Waring rank of a homogenous polynomial the smallest length of such a Waring decomposition. Apolarity theory provides a very powerful algebraic tool to study Waring decompositions of a homogeneous polynomial by studying sets of points apolar to the polynomial, i.e., sets of points whose defining ideal is contained in the so-called apolar ideal of the polynomial. In this talk, I want to introduce the concept of Waring locus of a homogeneous polynomial, i.e., the locus of linear forms which may appear in a minimal Waring decomposition. Then, after showing some example on how Waring loci can be computed in specific cases via apolarity theory. I explain how they may be used to construct minimal Waring decompositions. These results are from recent joint works with E. Carlini, M.V. Catalisano, and B. Mourrain.