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 $\left(\right(k+1)\times (k+1\left)\right)$-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 $d=d\left(k\right)$ 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 size 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 ways.