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We find conditions that guarantee that a decomposition of a genericthird-order tensor in a minimal number of rank-$1$ tensors (canonical polyadicdecomposition (CPD)) is unique up to permutation of rank-$1$ tensors. Then weconsider the case when the tensor and all its rank-$1$ terms have symmetricfrontal slices (INDSCAL). Our results complement the existing bounds forgeneric uniqueness of the CPD and relax the existing bounds for INDSCAL. Thederivation makes use of algebraic geometry. We stress the power of theunderlying concepts for proving generic properties in mathematical engineering.

Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL