Determinantal Conjecture

Definitions and mathematical statement

Using the conventional complex-matrix setting, let $A,B\in M_n({\mathbb{C}})$, where $n\geq1$ and $A^*$ denotes the conjugate transpose of $A$. A matrix $A$ is normal if $AA^*=A^*A$. The literal source request is: characterize $\det(A+B)$ when both $A$ and $B$ are normal. It specifies neither prescribed input data for the characterization nor an inequality, inclusion, or other proposed conclusion. Consequently this request alone is not a yes/no conjecture.

Separate candidate identified by the supplied review. The dated triage says that the likely intended target is the Marcus--de Oliveira convex-hull conjecture. To state that candidate explicitly, list the eigenvalues of $A$ and $B$, with algebraic multiplicities, as $\alpha_1,\ldots,\alpha_n$ and $\beta_1,\ldots,\beta_n$. Let $S_n$ be the group of permutations of $\{1,\ldots,n\}$, and set \[ z_\sigma=\prod_{i=1}^{n}(\alpha_i+\beta_{\sigma(i)})\qquad(\sigma\in S_n). \] For a finite set of complex numbers its convex hull is taken in the real plane ${\mathbb{C}}\cong{\mathbb{R}}^2$. The candidate assertion is that, for every $n\geq1$ and every pair of normal complex $n\times n$ matrices, \[ \det(A+B)\in\operatorname{conv}\{z_\sigma:\sigma\in S_n\}. \] Equivalently, there would exist real coefficients $t_\sigma\geq0$, depending on $A,B$, with $\sum_{\sigma\in S_n}t_\sigma=1$ and \[ \det(A+B)=\sum_{\sigma\in S_n}t_\sigma \prod_{i=1}^{n}(\alpha_i+\beta_{\sigma(i)}). \] This candidate is supplied for source review, not silently substituted for the imported request.

For a prescribed-spectra characterization one would instead have to specify the desired information about the set \[ \left\{\det\!\left(\operatorname{diag}(\alpha_1,\ldots,\alpha_n) +U\operatorname{diag}(\beta_1,\ldots,\beta_n)U^*\right): U\in M_n({\mathbb{C}}),\ U^*U=UU^*=I\right\}. \] The original request does not say whether it seeks this entire set, bounds on it, or the convex-hull inclusion above. In particular the candidate inclusion is not an assertion that the entire convex hull is attained.

The literature triage dated 2026-08-17 reports partial answers by Bebiano--Queiró for prescribed spectra, a further sufficient condition by Shitov in 2025, and a July 2026 variational treatment by Mulero-Martínez that describes the general normal case as open. Exact hypotheses of those partial results are not reproduced. The review therefore distinguishes the ill-specified literal record from the likely conjectural reconstruction. Its partially solved classification and the registry's open label are retained as supplied assessments rather than an independent status judgment.

Short English statement

The import asks only to characterize the determinant of a sum of normal matrices. Its review suggests the Marcus--de Oliveira conjecture: the determinant should lie in the convex hull of products formed by pairing the two eigenvalue lists. That precise candidate is stated separately because prescribed spectra, the convex hull, and the required conclusion are missing from the original wording.

Sources

• UnsolvedMath local registry, data/unsolvedmath/problems.json, permanent ID 1450, source code catalogue entry, "Determinantal Conjecture," including the literature triage dated 2026-08-17 and its explicit warning that the intended statement is missing.
• Natália Bebiano and João Filipe Queiró, "The determinant of the sum of two normal matrices with prescribed eigenvalues," Linear Algebra and its Applications 71 (1985), 23--28, https://doi.org/10.1016/0024-3795(85)90231-9. The supplied review cites its prescribed-spectrum formulation and partial answers.
• Yaroslav N. Shitov, "A further sufficient condition for the determinantal conjecture," Izvestiya: Mathematics 89 (2025), 862--869, https://www.mathnet.ru/eng/im9292. The supplied review cites a further sufficient condition for the convex-hull conjecture.
• Juan Ignacio Mulero-Martínez, "A variational framework for determinantal inequalities of normal matrices: Successes and obstructions," Linear Algebra and its Applications 740 (2026), 19--38, https://doi.org/10.1016/j.laa.2026.03.019. The review cites its precise conjecture statement and assessment that the general normal case remains open.
• Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems. Dataset curation and original metadata are supplied under CC BY 4.0; referenced source material retains its own terms. None of the linked references was independently consulted for this draft.