Abstract.
The rank of a symmetric tensor f is the minimal number of summands in an expression of f as a sum of symmetric tensors of rank one, or equivalently the homogeneous form f as a sum powers of linear forms. In the classical notion of apolarity a sum of powers decomposition of f corresponds to an finite apolar subscheme to f. Finite apolar subscheme to f may have shorter length than the rank. I shall discuss additive decompositions of f corresponding to apolar subschemes of minimal length, and present some joint work with F.O. Schreyer on the Hilbert scheme of apolar subschemes of minimal length for a quadratic form.