Tensor rank is a very classical notion, naturally arising in algebraic geometry, algebraic statistics, and complexity theory. In this context an old problem is to determine the rank of a given tensor, that is, to find defining equations for the variety of tensors of a given (border) rank k. In this talk I will report on joint work with Jan Draisma, where we prove qualitative results on the variety of p-tensors of border rank at most k.

For example we show that this variety is defined by equations of degree at most d = d(k), independent of the number of tensor factors (or the dimension of each factor).