On the tensor rank of $3\times 3$ permanent and determinant

Abstract

The tensor rank and border rank of the $3\times 3$ determinant tensor is known to be $5$ if characteristic is not two. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well. We also include an analysis of $5\times 5$ and $7\times 7$ determinant and permanent tensors, as well as the symmetric $3\times 3$ permanent and determinant tensors. We end with some remarks on binary tensors.