Border Waring Rank via Asymptotic Rank

Abstract

We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound extends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.

Apolarity, border rank and multigraded Hilbert scheme

“…We also calculate or provide new lower bounds for many other monomials (Examples 6.4-6.8), focusing on the Veronese case. Note that there is overlap between our results, and the claims of [Oedi19] and [CGO19], but these two articles have gaps as explained in §6.1.…”

“…Both rank r P n (F ) and the variety of sums of powers V SP (F, r X (F )) are calculated in [CCG12] and in [BBT13]. Moreover, the border rank br P n (F ) is discussed in [Oedi19] and [CGO19], however both methods have gaps at the time of submission of this article. For some other low dimensional toric varieties X, the rank of some monomials is calculated and estimated in [Gałą14].…”

“…The approach of Christandl, Gesmundo, and Oneto via asymptotic rank. The gap described in the previous paragraph affects the first version of [CGO19] 2 . There the authors claim to prove that the equality holds in (6.1) for all monomials.…”

Geometric conditions for strict submultiplicativity of rank and border rank

“…However, both inequalities in (1) can be strict in general, as shown in [CJZ18] for rank and [CGJ19] for border rank. Nonetheless, in [CGO19] it is shown that multiplicative lower bounds for rank extend to border rank and this fact is applied to the Ranestad-Schreyer method from [RS11] to compute the border rank of monomials.…”