Symmetrization maps and minimal border rank Comon’s conjecture
Abstract
Abstract One of the fundamental open problems in the field of tensors is the border Comon’s conjecture : given a symmetric tensor $$F\in (\mathbb {C}^n)^{\otimes d}$$ F ∈ ( C n ) ⊗ d for $$d\ge 3$$ d ≥ 3 , its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $$(\mathbb {C}^n)^{\otimes d}$$ ( C n ) ⊗ d of border rank n , i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $$n\le d+1$$ n ≤ d + 1 . Our technical tools are border apolarity and border varieties of sums of powers.
Fields of Study & Subjects
Advanced Topics in Algebra
Homotopy and Cohomology in Algebraic Topology
Advanced Topology and Set Theory
Publication details
- Journal / source
- Selecta Mathematica
- Publisher
- Springer Science and Business Media LLC
- Volume
- 32
- Issue
- 4
- Publication date
- 13 August 2026
- Language
- EN