  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.
