The border rank of the $4 \times 4$ determinant tensor is twelve

We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.

Publication Details

Journal
ArXiv.org
Published
2025-10-13
Primary Topic
Tensor decomposition and applications
Type
article
Field-Weighted Citation Impact
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article

The border rank of the $4 \times 4$ determinant tensor is twelve

ArXiv.org
Tensor decomposition and applications
article

The border rank of the $4 \times 4$ determinant tensor is twelve

article en

Abstract

We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.

ArXiv.org
National Research Foundation, National Research Foundation of Korea, Korea Advanced Institute of Science and Technology, Ministry of Science and ICT, South Korea
Reduced inequalities
Openalex Percentile: Top 50%
Tensor decomposition and applications
Open Access PDFOpenAlex
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The border rank of the $4 \times 4$ determinant tensor is twelve · ArXiv.org (2025) | TGRS Research Map | TGRS