  In this paper we study the locus of singular tuples of a complex valued
multisymmetric tensor. The main problem that we focus on is: given the set of
singular tuples of some general tensor, which are all the tensors that admit
those same singular tuples. Assume that the triangular inequality holds, that
is exactly the condition such that the dual variety to the Segre-Veronese
variety is an hypersurface, or equivalently, the hyperdeterminant exists. We
show in such case that, when at least one component has degree odd, this tensor
is projectively unique. On the other hand, if all the degrees are even, the
fiber is an $1$-dimensional space.
