The rank of the $5\times 5$ permanent tensor is sixteen

Abstract

In this paper, we show that the tensor rank of the $5\times 5$ permanent tensor is exactly $16$ over fields of characteristic zero, by proving the lower bound matching the known upper bound. The $5\times 5$ permanent tensor is a symmetric tensor corresponding to the monomial $x_1x_2x_3x_4x_5$, whose Waring rank is known to be $16$. Previously, it was known, by the higher-order Koszul flattening and Fischer's formula, that its tensor rank is either $15$ or $16$.