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

TL;DR

This paper proves that the tensor rank of the $3\times 3$ determinant tensor is 5 over fields of any characteristic, and provides analysis of larger and symmetric tensors, with remarks on binary tensors.

Contribution

It establishes the tensor rank of the $3\times 3$ determinant tensor as 5 in all characteristics and analyzes related larger and symmetric tensors.

Findings

Tensor rank of $3\times 3$ determinant tensor is 5 in all characteristics

Analysis of $5\times 5$ and $7\times 7$ determinant and permanent tensors

Remarks on binary tensors

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.