A 3× 3 counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem

@misc{shi2026counterexample,
  title = {{A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem}},
  author = {Shuo Shi and Juan Zhang and Yun Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07113},
  eprint = {2609.07113},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07113},
}

TY  - UNPB
TI  - A 3× 3 counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem
AU  - Shuo Shi
AU  - Juan Zhang
AU  - Yun Zhang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07113
UR  - https://arxiv.org/abs/2609.07113
AB  - We give a 3× 3 counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441--443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than 3. Hence the example is dimensionally minimal over the complex field.
ER  -