The border rank of the $4 \times 4$ determinant tensor is twelve

Abstract

We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczy\'nska-Buczy\'nski and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.