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Let T be a general complex tensor of format $(n_1,...,n_d)$. When thefraction $\prod_in_i/[1+\sum_i(n_i-1)]$ is an integer, and a natural inequality(called balancedness) is satisfied, it is expected that T has finitely manyminimal decomposition as a sum of decomposable tensors. We show how homotopytechniques allow us to find all the decompositions of T, starting from a givenone. Computationally, this gives a guess regarding the total number of suchdecompositions. This guess matches exactly with all cases previously known, andpredicts several unknown cases. Some surprising experiments yielded two newcases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have aunique decomposition as the sum of 6 and 4 decomposable tensors, respectively.We conjecture that these two cases together with the classically known matrixpencils are the only cases where generic identifiability holds, i.e., the onlyidentifiable cases. Building on the computational experiments, we use algebraicgeometry to prove these two new cases are indeed generically identifiable.

Homotopy techniques for tensor decomposition and perfect identifiability