A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks

TL;DR

This paper introduces a new explicit formula for the determinant with fewer terms, providing tighter bounds on tensor and Waring ranks, and revealing new geometric structures.

Contribution

It presents a novel explicit determinant formula, improves bounds on tensor and Waring ranks, and uncovers new tiling polytopes in high-dimensional spaces.

Findings

Determinant formula with superexponentially fewer terms

Tensor rank of $n imes n$ determinant bounded by Bell number

Exact tensor rank of $4 imes 4$ determinant over $ extbf{F}_2$ is 12

Abstract

We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over $\mathbb{F}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile $\mathbb{R}^n$.