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A 27 x 27 x 27 counterexample to Comon's conjecture

  • Benjamin Lovitz
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Abstract

We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.

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Cite

BibTeXRIS
BibTeX
@misc{lovitz2026counterexample,
  title = {{A 27 x 27 x 27 counterexample to Comon's conjecture}},
  author = {Benjamin Lovitz},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28292},
  eprint = {2609.28292},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28292},
}
RIS
TY  - UNPB
TI  - A 27 x 27 x 27 counterexample to Comon's conjecture
AU  - Benjamin Lovitz
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28292
UR  - https://arxiv.org/abs/2609.28292
AB  - We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.
ER  -