Tensor slice rank and Cayley's first hyperdeterminant

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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.