Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

Abstract

We show that if a homogeneous polynomial f in n variables has Waring rank $$n+1$$ , then the corresponding projective hypersurface $$f=0$$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of f. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.