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-by-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 ≥ 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-by-4 determinant over F₂ 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 ≥ 17, and lead to a new family of axis-aligned polytopes that tile Rⁿ.

Authors:

• Robin Houston
• Adam P. Goucher
• Nathaniel Johnston

Download:

• Official publication from Combinatorics, Probability and Computing
• Local preprint
• Preprint from arXiv:2301.06586 [math.CO]
• Slideshow presentation

Cite as:

• R. Houston, A. P. Goucher, and N. Johnston. A new formula for the determinant and bounds on its tensor and Waring ranks. Combinatorics, Probability and Computing, 33(6):769–794, 2024.

Supplementary Material:

• det4f2 – code for assisting in the proof that the 4-by-4 determinant over the field of 2 elements has tensor rank 12