On generic identifiability of symmetric tensors of subgeneric rank

Abstract

We prove that the general symmetric tensor in S d C n + 1 S^d\mathbb {C}^{n+1} of rank r r is identifiable, provided that r r is smaller than the generic rank. That is, its Waring decomposition as a sum of r r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ( d = 3 d=3 ), while for d ≥ 4 d\ge 4 we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.