Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula

In this paper, we improve the upper bound for the tensor rank of the $n\times n$ determinant tensor to ${\mathcal{B}}_n-2^{n-1}+n+1$ by finding a new tensor decomposition of the tensor, where ${\mathcal{B}}_n$ denotes the $n$-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston–Goucher–Johnston with a tensor that is essentially the same as the $n\times n$ permanent tensor, and to use Glynn’s formula. This upper bound gives an improved upper bound for the Waring rank of the $n\times n$ determinant polynomial for $n\ge 17$ as well.