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