Product Ranks of the 3×3 Determinant and Permanent

Nathan Ilten; Zach Teitler

doi:10.4153/CMB-2015-076-1

Product Ranks of the 3×3 Determinant and Permanent

Nathan Ilten, Zach Teitler

Mathematics Department

Simon Fraser University

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We show that the product rank of the 3×3 determinant det₃ is 5, and the product rank of the 3×3 permanent perm₃ is 4. As a corollary, we obtain that the tensor rank of det₃ is 5 and the tensor rank of perm₃ is 4. We show moreover that the border product rank of perm_n is larger than n for any n ≥ 3.

Original language	English
Pages (from-to)	311-319
Number of pages	9
Journal	Canadian Mathematical Bulletin
Volume	59
Issue number	2
DOIs	https://doi.org/10.4153/CMB-2015-076-1
State	Published - Jun 2016

Keywords

Determinant
Fano schemes
Permanent
Product rank
Tensor rank

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title = "Product Ranks of the 3×3 Determinant and Permanent",

abstract = "We show that the product rank of the 3×3 determinant det3 is 5, and the product rank of the 3×3 permanent perm3 is 4. As a corollary, we obtain that the tensor rank of det3 is 5 and the tensor rank of perm3 is 4. We show moreover that the border product rank of permn is larger than n for any n ≥ 3.",

keywords = "Determinant, Fano schemes, Permanent, Product rank, Tensor rank",

author = "Nathan Ilten and Zach Teitler",

note = "Publisher Copyright: {\textcopyright} Canadian Mathematical Society 2016.",

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AU - Teitler, Zach

N1 - Publisher Copyright: © Canadian Mathematical Society 2016.

PY - 2016/6

Y1 - 2016/6

N2 - We show that the product rank of the 3×3 determinant det3 is 5, and the product rank of the 3×3 permanent perm3 is 4. As a corollary, we obtain that the tensor rank of det3 is 5 and the tensor rank of perm3 is 4. We show moreover that the border product rank of permn is larger than n for any n ≥ 3.

AB - We show that the product rank of the 3×3 determinant det3 is 5, and the product rank of the 3×3 permanent perm3 is 4. As a corollary, we obtain that the tensor rank of det3 is 5 and the tensor rank of perm3 is 4. We show moreover that the border product rank of permn is larger than n for any n ≥ 3.

KW - Determinant

KW - Fano schemes

KW - Permanent

KW - Product rank

KW - Tensor rank

UR - http://www.scopus.com/inward/record.url?scp=84978971909&partnerID=8YFLogxK

U2 - 10.4153/CMB-2015-076-1

DO - 10.4153/CMB-2015-076-1

M3 - Article

AN - SCOPUS:84978971909

SN - 0008-4395

VL - 59

SP - 311

EP - 319

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

IS - 2

ER -

Product Ranks of the 3×3 Determinant and Permanent

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Keywords

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