About the real Waring rank of polynomials and their geometry

Abstract

The Waring rank of the given polynomial is the minimal number of linear forms whose sum of powers is equal to the polynomial. We study real ternary and quaternary forms whose real rank equals the generic complex rank,and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for ternary quadrics and cubics. For ternary quintics and quaternary cubic we determine the real rank boundary.For ternary quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants. For the quaternary case, we also obtain complete results for quadrics and cubics, and partial results for quartics. Also we present some algorithms to calculate the semialgebraic set of sums of powers and real rank boundaries.