  We introduce an elementary method to study the border rank of polynomials and
tensors, analogous to the apolarity lemma. This can be used to describe the
border rank of all cases uniformly, including those very special ones that
resisted a systematic approach. We also define a border rank version of the
variety of sums of powers and analyse its usefulness in studying tensors and
polynomials with large symmetries. In particular, it can be applied to provide
lower bounds for the border rank of some very interesting tensors, such as the
matrix multiplication tensor. We work in a general setting, where the base
variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth
toric projective variety. A critical ingredient of our work is an irreducible
component of a multigraded Hilbert scheme related to the toric variety in
question.
