Hadamard's maximal determinant problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For every positive integer $n$, let \[D(n)=\max\{|\det A|:A\in\{1,-1\}^{n\times n}\}. \] The maximum exists because the set of admissible matrices is finite.

Statement recorded in the supplied source.

Determine $D(n)$ exactly for every $n\geq1$. [S2]

Short English statement

Find the largest absolute determinant among all square sign matrices of every order.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1600016 (AMR-015-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 P. Browne, R. Egan, F. Hegarty and P. Ó Catháin, A Survey of the Hadamard Maximal Determinant Problem, arXiv:2104.06756v4 (2021), introduction and §2. https://arxiv.org/abs/2104.06756.