Tensor slice rank and Cayley's first hyperdeterminant

2021/07/19 by Amanov, Alimzhan, Yeliussizov, Damir · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.08864

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.

Cited by

• New Identity for Cayley's First Hyperdeterminant with Applications to Symmetric Tensors and Entanglement
• Isotropy and completeness indices of multilinear maps
• Triangular tensors and set-intersection problems

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• A gap in the slice rank of k-tensors
• The slice rank of a direct sum
• Partition Rank and Partition Lattices
• Bounds on the tensor rank
• A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction