  Kruskal's theorem states that a sum of product tensors constitutes a unique
tensor rank decomposition if the so-called k-ranks of the product tensors are
large. In this work, we prove a result in which the k-rank condition of
Kruskal's theorem is weakened to the standard notion of rank, and the
conclusion is relaxed to a statement on the linear dependence of the product
tensors. Our result implies a generalization of Kruskal's theorem. Several
adaptations and generalizations of Kruskal's theorem have already been
obtained, but most of these results still cannot certify uniqueness when the
k-ranks are below a certain threshold. Our generalization contains several of
these results, and can certify uniqueness below this threshold. As a corollary
to our results, we prove that if n product tensors form a circuit, then they
have rank greater than one in at most n-2 subsystems. This generalizes several
recent results in this direction, and is sharp.
