Geodesic
Product Ranks of the 3 × 3 Determinant and Permanent
Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 311-319

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DOI

We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a corollary, we obtain that the tensor rank of ${{\det }_{3}}$ is $5$ and the tensor rank of $\text{per}{{\text{m}}_{3}}$ is $4$ . We show moreover that the border product rank of $\text{per}{{\text{m}}_{3}}$ is larger than $n$ for any $n\,\ge \,3$ .
DOI : 10.4153/CMB-2015-076-1
Mots-clés : 15A21, 15A69, 14M12, 14N15, product rank, tensor rank, determinant, permanent, Fano schemes 
Ilten, Nathan; Teitler, Zach. Product Ranks of the 3 × 3 Determinant and Permanent. Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 311-319. doi: 10.4153/CMB-2015-076-1
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     title = {Product {Ranks} of the 3 {\texttimes} 3 {Determinant} and {Permanent}},
     journal = {Canadian mathematical bulletin},
     pages = {311--319},
     year = {2016},
     volume = {59},
     number = {2},
     doi = {10.4153/CMB-2015-076-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-076-1/}
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