  Cayley's first hyperdeterminant is a straightforward generalization of
determinants for tensors. We prove that nonzero hyperdeterminants imply lower
bounds on some types of tensor ranks. This result applies to the slice rank
introduced by Tao and more generally to partition ranks introduced by Naslund.
As an application, we show upper bounds on some generalizations of colored
sum-free sets based on constraints related to order polytopes.
