Product ranks of the \(3\times 3\) determinant and permanent (Q2810709)

The homogeneous polynomials \(\mathrm{perm}_n\) and \(\det_n\) are the permanent and the determinant, respectively, of an \(n\times n\) matrix whose entries are \(n^2\) different variables. The \textit{product rank} \(\mathrm{pr}(F)\) of a homogeneous form \(F\) of degree \(d\) is the smallest integer \(r\) such that there exist homogeneous linear forms \(l_{ij}\) with NEWLINE\[NEWLINEF = \sum_{i=1}^r \prod_{j=1}^d l_{ij}.NEWLINE\]NEWLINE It has been known that \(3 \leq \mathrm{pr}(\mathrm{perm}_3) \leq 4\) and \(4 \leq \mathrm{pr}(\det_3) \leq 5\). The authors show that \(\mathrm{pr}(\mathrm{perm}_3) = 4\) and \(\mathrm{pr}(\det_3) = 5\).