The border rank of the $4 \times 4$ determinant tensor is twelve

TL;DR

The paper determines the border rank of the $4\times 4$ determinant tensor as $12$ by applying border apolarity and the fixed ideal framework of Conner, Harper, and Landsberg. It refines the order-4 border-rank criterion with a sequence of $\mathbb{B}$-fixed tests, identifies a unique $\mathbb{B}$-fixed candidate, and uses a crucial $(1111)$-test to show that no border-rank-11 decomposition exists. Consequently, $\underline{\mathbf{R}}(\operatorname{det}_4)=12$ over $\mathbb{C}$, matching the known upper bound and thus exactly pinning the border rank. The work also discusses the Fixed Ideal Theorem’s limitations, illustrating that for some tensors (notably small determinants) the method may not yield a border-rank decomposition, and highlights open questions for higher-dimensional determinants such as $\det_n$ with $n\ge 5$. These insights advance geometric complexity theory by clarifying how algebro-geometric tools constrain tensor decompositions and by outlining computational approaches (e.g., Macaulay2) used in the verification.

Abstract

We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.

Theorems & Definitions (13)

• Theorem 1.1
• Definition 2.1: Concise tensor
• Definition 2.2: Secant variety and border $X$-rank
• Definition 2.3: Border rank
• Definition 2.4: Annihilator
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8: Weak border apolarity theorem, MR4332674
• Theorem 2.9: Fixed ideal theorem, MR4332674