Bounded-rank tensors are defined in bounded degree

TL;DR

This paper establishes that tensors with bounded border rank are characterized by polynomial equations of uniformly bounded degree, extending the matrix rank polynomial characterization to higher-dimensional tensors.

Contribution

It proves the existence of a uniform polynomial degree bound for defining equations of tensors with bounded border rank across all dimensions.

Findings

Tensors of border rank at most k are defined by polynomials of degree at most d(k).

The degree bound d(k) is independent of tensor dimension and size.

The proof uses an infinite-dimensional limit and symmetry properties.

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.