Border Ranks of Monomials

Understand

Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and their non-vanishing provides lower bounds for border ranks of tensors and in particular polynomials.

• We study monomial-optimal shapes for Young flattenings, which exhibit the limits of the Young flattening method.
• In particular, they provide the best possible lower bound for large classes of monomials including all monomials up to degree 6, monomials in 3 variables, and any power of the product of variables.
• On the other hand, for degree 7 and higher there are monomials for which no Young flattening can give a lower bound that matches the conjecturally tight upper bound of Landsberg and Teitler.