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

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 $$\Delta (T):=s(T)-r(T)\le \min \{C_{ec}(e(T)),C_{dc}(d(T))\}.$$ The excess-controlled function $C_{ec}(e)$ is zero for $0\le e\le 3$ and equals $9e-3$ for $e\ge 4$. In particular, the equality $C_{ec}(e(T))=0$ for $0\le e\le 3$ 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 $1\le d\le 18$, and equals $d^2/4+d/4+3/2$ for $d\ge 19$. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range $e\le 3$. 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.

arXiv comment30 pages