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PreprintCC BY-NC-ND 4.0arXiv

The border Waring rank of x1⋯ xn is 2n−1

  • Jong In Han
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Abstract

In this paper, we show that the border Waring rank of x1x2⋯ xn over fields of characteristic zero is exactly 2n−1. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the n× n permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the n× n determinant tensor, which is not symmetric.

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Cite

BibTeXRIS
BibTeX
@misc{han2026border,
  title = {{The border Waring rank of $x_1\cdots x_n$ is $2^n-1$}},
  author = {Jong In Han},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09141},
  eprint = {2609.09141},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09141},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The border Waring rank of x_1⋯ x_n is 2^n−1
AU  - Jong In Han
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09141
UR  - https://arxiv.org/abs/2609.09141
AB  - In this paper, we show that the border Waring rank of x_1x_2⋯ x_n over fields of characteristic zero is exactly 2^n−1. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the n× n permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the n× n determinant tensor, which is not symmetric.
ER  -