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 $F\in ({\mathbb{C}}^n{)}^{\otimes d}$ for $d\ge 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $({\mathbb{C}}^n{)}^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\le d+1$. Our technical tools are border apolarity and border varieties of sums of powers.