Tensor slice rank and Cayley's first hyperdeterminant

TL;DR

This paper explores the properties of Cayley's first hyperdeterminant, demonstrating its implications for tensor ranks and applying these findings to bounds on generalized sum-free sets.

Contribution

It establishes a link between nonzero hyperdeterminants and lower bounds on tensor ranks, extending to partition ranks and applications in combinatorial bounds.

Findings

Nonzero hyperdeterminants imply lower bounds on tensor ranks.

Results apply to slice rank and partition ranks.

Provides upper bounds on generalized sum-free sets.

Abstract

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.