Product Ranks of the $3\times 3$ Determinant and Permanent

Abstract

We show that the product rank of the $3\times 3$ determinant is $5$, and the product rank of the $3\times 3$ permanent is $4$. As a corollary, we obtain that the tensor ranks of the $3 \times 3$ determinant and permanent are $5$ and $4$, respectively. We show moreover that the border product rank of the $n \times n$ permanent is larger than $n$ for any $n\geq 3$.