Border Ranks of Monomials

Abstract

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.

Sufficient conditions for Strassen’s additivity conjecture

“…However, Oeding has very recently shown that brk(M) = (a 1 + 1) • • • (a n−1 + 1) for every monomial M [20]. While it is not explicitly stated in [20], it seems to be the case that Oeding's technique extends to show that if F 1 , . .…”

“…The Waring ranks of monomials and sums of monomials in independent variables are known [8], cactus ranks of monomials are known [23], and border ranks of monomials have been found recently as well [20]. However cactus ranks and border ranks of sums of monomials in independent variables are not known.…”

Border Waring Rank via Asymptotic Rank

“…The upper bound R d (m) ≤ 1 an+1 n i=0 (a i + 1) was proved by Landsberg and Teitler in [LT10, Theorem 11.2] and equality was shown under the assumption a n ≥ a 0 +• • •+a n−1 . In [Oed16], Oeding proved equality for a number of other families of monomials. These results achieved the desired lower bound via flattening methods.…”

“…Similarly every tensor for which the cactus rank is multiplicative under tensor product satisfies (⋆ c ) together with all its tensor powers. In particular all monomials for which the flattening lower bounds from [Oed16] hold and all the tensor products of any number of them satisfy (⋆).…”

Schur Polynomials do not have small formulas if the Determinant doesn't!

“…Further, via reductions, the elementary and power symmetric polynomials have been used to define restricted models of algebraic computation known as the symmetric circuit model [Shp01] and the Σ ∧ Σ model (or Waring rank), which in turn have been significantly investigated (see, e.g. [Sax08,LT10,Oed16]).…”