A new formula for the determinant and bounds on its tensor and Waring ranks

2023/01/16 by Robin Houston, Adam P. Goucher, Nathaniel Johnston · 2 voices · 2 citations
Mathematics · #Tensor decomposition and applications #Advanced Optimization Algorithms Research

paper · pdf · doi:10.1017/s0963548324000233

Abstract

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 × 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 × 4 determinant over \mathbbF₂ 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 .

Cited by

• The border rank of the 4 × 4 determinant tensor is twelve
• Tensor rank of the determinant and periodic triangulations of ℝⁿ

Discussions

• A New Formula for the Determinant [hn, 95 points, 33 comments]
• If you just consider lattices where there are no gaps (but there usually are overlaps), then you cut out parts of the brick to eliminate overlaps, and calculate the volume of the remaining shape, this [bsky, 2 points, 1 comments]

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