The border Waring rank of $x_1\cdots x_n$ is $2^{n-1}$

Abstract

In this paper, we show that the border Waring rank of $x_1{x}_2\cdots x_n$ over fields of characteristic zero is exactly $2^{n-1}$. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the $n\times n$ permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the $n\times n$ determinant tensor, which is not symmetric.

arXiv comment17 pages