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

TL;DR

This paper determines the product rank of the 3x3 determinant and permanent, revealing their tensor ranks and establishing that the border product rank of the n×n permanent exceeds n for all n≥3.

Contribution

It provides exact product ranks for the 3x3 determinant and permanent, and shows the border product rank of the n×n permanent surpasses n for all larger n.

Findings

Product rank of 3x3 determinant is 5.

Product rank of 3x3 permanent is 4.

Border product rank of n×n permanent exceeds n for n≥3.

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 $n\geq 3$.