Apolarity, border rank, and multigraded Hilbert scheme

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