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 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