Bounded-rank tensors are defined in bounded degree

Abstract

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d = d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its sizes in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial way.

Corrected the formulation of Corollary 1.4---thanks to Sonja Petrovic!

References in corpus (3)

• Identifiability of parameters in latent structure models with many observed variables
• Syzygies of Segre embeddings and Delta-modules
• Toward a salmon conjecture

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