Symmetrization maps and minimal border rank Comon’s conjecture

One of the fundamental open problems in the field of tensors is the border Comon’s conjecture : given a symmetric tensor F∈ (Cⁿ)⊗ d F ∈ ( C n ) ⊗ d for d≥ 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 (Cⁿ)⊗ 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≤ d+1 n ≤ d + 1 . Our technical tools are border apolarity and border varieties of sums of powers.