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Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula

  • Jeong-Hoon Ju
  • Hayoung Choi
  • Taehyeong Kim
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Abstract

In this paper, we improve the upper bound for the tensor rank of the n × n determinant tensor to ℬn−2n−1+n+1 by finding a new tensor decomposition of the tensor, where ℬn denotes the n-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston--Goucher--Johnston with a tensor that is essentially the same as the n × n permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the n × n determinant polynomial for n ≥ 17 as well.

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Cite

BibTeXRIS
BibTeX
@misc{ju2026improved,
  title = {{Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula}},
  author = {Jeong-Hoon Ju and Hayoung Choi and Taehyeong Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18394},
  eprint = {2610.18394},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18394},
}
RIS
TY  - UNPB
TI  - Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula
AU  - Jeong-Hoon Ju
AU  - Hayoung Choi
AU  - Taehyeong Kim
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18394
UR  - https://arxiv.org/abs/2610.18394
AB  - In this paper, we improve the upper bound for the tensor rank of the n × n determinant tensor to ℬ_n−2^n−1+n+1 by finding a new tensor decomposition of the tensor, where ℬ_n denotes the n-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston--Goucher--Johnston with a tensor that is essentially the same as the n × n permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the n × n determinant polynomial for n ≥ 17 as well.
ER  -