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

TL;DR

This work introduces a new explicit determinant formula with $B_n$ terms, where $B_n$ is the $n$-th Bell number, achieving far fewer terms than the classical Leibniz expansion and subsuming the $n=3$ five-term identity. From this formula, the authors derive field-independent upper bounds on the tensor and Waring ranks of the determinant: $\operatorname{Trank}(\operatorname{det}^n_{\mathbb{F}})\le B_n$ and $\operatorname{Wrank}(\operatorname{det}^n_{\mathbb{F}})\le 2^{n-1} B_n$, with tighter bounds in positive characteristic (e.g., $\operatorname{Trank}(\operatorname{det}^n_{\mathbb{F}})\le 2^n- n$ in characteristic $2$) and an exact result $\operatorname{Trank}(\det_4^{\mathbb{F}_2})=12$. The authors provide two independent proofs of the formula—a combinatorial one and a geometric one—alongside a geometric tiling interpretation that gives rise to axis-aligned polytope tilings and a flip-graph structure on ordered partial partitions. They also obtain refined bounds for the Waring rank and tensor rank over finite fields and demonstrate optimality in small cases, notably establishing the exact rank 12 for $4\times4$ determinant over $\mathbb{F}_2$. These results advance the understanding of determinant representations, connecting combinatorial partitions, polytope tilings, and algebraic complexity in a unified framework with potential implications for algebraic complexity and computational algebra.

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 $\mathbb{R}^n$.

Figures (5)

• Figure 1: Two tilings of $\mathbb{R}^2$ on the same lattice in which the tiles have area equal to $\operatorname{det}\left(\left[5-2-13\right]\right) = 13$.
• Figure 2: Two $5 \times 3$ rectangular not-quite-tilings of $\mathbb{R}^2$ coming from matrices with diagonal entries $5$ and $3$. The shaded rectangle is $C_A$, while the other rectangles are its translates on the lattice $\Lambda_A$.
• Figure 3: A summary of the relationship between the properties $P_1(A)$--$P_5(A)$. Any two of the properties in the left Y-shape imply the third, and any two of the properties in the right Y-shape imply the third.
• Figure 4: The polytope $F_B$ for the matrix $B = (n+1)I - J$, with $n = 3$. Its volume is equal to the sum and difference of the volumes of $5$ different cubes, corresponding to the $5$-term formula for the determinant \ref{['eq:det_3x3']}.
• Figure 5: The $1$-skeleton of a polytope is the set of its vertices and $1$-dimensional edges between them. Propositions \ref{['prop:FA_vertices']} and \ref{['prop:FA_edges']} show that the $1$-skeleton of the polytope $F_A$ is isomorphic to a graph whose vertices are the ordered partial partitions on $[n]$ and whose edges are described by the involutions $f_z$.

Theorems & Definitions (29)

• Lemma 1
• Theorem 2
• Corollary 3
• Lemma 4
• proof
• Lemma 5
• proof : Proof of Lemma \ref{['lem:FA_overlaps']}
• Lemma 6
• proof
• Lemma 7