Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors

@misc{li2026upper,
  title = {{Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors}},
  author = {Jianze Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14002},
  eprint = {2610.14002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14002},
}

TY  - UNPB
TI  - Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors
AU  - Jianze Li
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.14002
UR  - https://arxiv.org/abs/2610.14002
AB  - Let T be a nonzero real symmetric third-order tensor. Denote by r(T) its CP rank, by s(T) its symmetric rank, by d(T) its essential dimension, and put e(T)=r(T)−d(T). We establish two complementary upper bounds for the Comon rank gap Δ(T):= s(T)−r(T)≤min{C_ec(e(T)), C_dc(d(T))}. The excess-controlled function C_ec(e) is zero for 0≤ e≤3 and equals 9e−3 for e≥4. In particular, the equality C_ec(e(T))=0 for 0≤ e≤3 extends Friedland's flattening-rank criterion for real symmetric third-order tensors by two additional rank layers. The dimension-controlled function C_dc(d) is zero for d=0, equals (d^2−4d+10)/3 for 1≤ d≤18, and equals d^2/4+d/4+3/2 for d≥19. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range e≤3. The dimension estimate combines a transverse polar-completion bound with a shifted estimate for the maximum real Waring rank. Thus the two bounds measure different features of the same tensor and remain effective in complementary regimes.
ER  -