Spaces of sums of powers and real rank boundaries

Abstract

We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of mathrm{SSP} to various loci in the Hilbert scheme of four points in mathbb {P}^3. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary cubics the boundary of this locus is a degree forty hypersurface J(sigma _3(v_3(mathbb {P}^3)),tau (v_3(mathbb {P}^3))).