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The Comon Rank Gap of a Third-Order Symmetric Tensor Can Exceed One
Abstract
For a third-order symmetric tensor, the Comon rank gap is the difference between its CP rank and its symmetric rank. We study direct sums of Lovitz's rational $27$-dimensional tensor, whose two ranks are $55$ and $56$. Two copies have ranks $110$ and $112$, and three copies have ranks $165$ and $168$, over both the real and the complex fields. Thus direct sums yield concise cubic tensors on spaces of dimensions $54$ and $81$ with Comon rank gaps two and three.
Related
- Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors
- Upper Bounds on the Strassen CP Rank Gap and Symmetric Rank Gap of Third-Order Tensors
- Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization
- The third symmetric product of a set is a set in HoTT
- SL(n)-covariant symmetric tensor valuations on polytopes
BibTeX
@misc{li2026comon,
title = {{The Comon Rank Gap of a Third-Order Symmetric Tensor Can Exceed One}},
author = {Jianze Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01660},
eprint = {2610.01660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01660},
}
RIS
TY - UNPB
TI - The Comon Rank Gap of a Third-Order Symmetric Tensor Can Exceed One
AU - Jianze Li
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01660
UR - https://arxiv.org/abs/2610.01660
AB - For a third-order symmetric tensor, the Comon rank gap is the difference between its CP rank and its symmetric rank. We study direct sums of Lovitz's rational $27$-dimensional tensor, whose two ranks are $55$ and $56$. Two copies have ranks $110$ and $112$, and three copies have ranks $165$ and $168$, over both the real and the complex fields. Thus direct sums yield concise cubic tensors on spaces of dimensions $54$ and $81$ with Comon rank gaps two and three.
ER -