Isotropic rank of harmonic polynomials

The Waring problem consists in computing the Waring rank of a homogeneous polynomial, i.e. the minimum number of powers of linear forms of which it is the sum. It is an NP-problem and we have only partial solutions. In particular, we can determine the rank of general forms, quadrics, monomials, binary forms, ternary cubics and a few other examples. In this talk we restrict to the class of harmonic polynomials, i.e. polynomials which are in the kernel of the Laplace operator, and we define the isotropic rank of a harmonic polynomial as the minimum number of powers of isotropic linear forms of which it is the sum. For the study of this rank, we introduce harmonic apolarity theory and the needed geometrical dictionary, and we use them to recover all the analogous results known for the Waring problem. In particular, we determine the isotropic rank for general harmonic forms, harmonic quadrics, harmonic monomials and harmonic ternary forms. This is a joint work with C. Flavi.