We study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus is: given the set of singular tuples of a general tensor, which are all the tensors that admit those same singular tuples. Assume that the triangular inequality holds, we show that when some 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.
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