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arXiv · 2610.14002

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

Jianze Li
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Abstract

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 $$Δ(T):= s(T)-r(T)\leq\min\{C_{ec}(e(T)),\ C_{dc}(d(T))\}.$$ The excess-controlled function $C_{ec}(e)$ is zero for $0\leq e\leq3$ and equals $9e-3$ for $e\geq4$. In particular, the equality $C_{ec}(e(T))=0$ for $0\leq e\leq3$ 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 $1\leq d\leq18$, and equals $d^2/4+d/4+3/2$ for $d\geq19$. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range $e\leq3$. 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.

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Jianze Li. 2026-09-27. Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors. https://arxiv.org/abs/2610.14002

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