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

On tensors that are determined by their singular tuples