Symmetrization maps and minimal border rank Comon's conjecture

2024/11/08 by Mańdziuk, Tomasz, Ventura, Emanuele
#14C05 #68Q17 #Algebraic Geometry (math.AG) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Primary: 14N07 #Secondary: 15A69

paper · doi:10.48550/arxiv.2411.05721

Abstract

One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor F∈(ℂⁿ)^{⊗ d} for d≥ 3, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in (ℂⁿ)^{⊗ d} of border rank n, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever n≤ d+1. Our technical tools are border apolarity and border varieties of sums of powers.

Related

• On Comon's and Strassen's conjectures
• Concise tensors of minimal border rank
• Apolarity, border rank and multigraded Hilbert scheme
• Classification and degenerations of small minimal border rank tensors via modules
• Explicit tensors of border rank at least 2n-1