Border rank of monomials via asymptotic rank

The (symmetric) rank of a (symmetric) tensor is the smallest length of an expression of the tensor as a linear combination of (symmetric) decomposable tensors. The border rank of a (symmetric) tensor is the smallest (symmetric) rank of the elements of a one-parameter family of (symmetric) tensors whose limit is equal to the given tensor. Computing rank and border rank of an explicit tensor can be a very difficult task. Upper bounds are often found by providing explicit expressions of the tensor, but lower bounds require theoretical arguments and are usually more difficult to find. In the case of border rank, a lower bound is given by the so-called asymptotic rank of a tensor which measure the growth of the rank when considering tensor-powers of the tensor. This notion, which can be connected to classic works by Strassen, was recently defined in a paper by Christandl, Gesmundo and Jensen.

In this talk, I want to show how algebraic tools from apolarity theory can be used to compute these different notions of rank in the case of symmetric tensors, i.e., homogeneous polynomials. In particular, I will show how to apply these tools in the case of monomials. This talk is based on the recent pre-print arXiv:1907.03487 with Matthias Christandl and Fulvio Gesmundo (U. of Copenaghen).