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Determinantal conjecture: determinantal_conjecture

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Added 1 Nov 2026 · not yet checked against the literature · importance 2.70

What the prompt says

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Statement

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod\_i (\lambda(A)\_i + \lambda(B)\_{\sigma(i)})$ ? Here the numbers $\lambda(A)\_i$ and $\lambda(B)\_i$ are the eigenvalues of $A$ and $B$ , and $\sigma$ is an element of the symmetric group $S\_n$ .

Lean statement

theorem determinantal_conjecture
    (n : Type) [Fintype n] [DecidableEq n]
    (d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :
    (U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det
      ∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Perm n } := by
  sorry

Sources(2)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Wikipedia/DeterminantalConjecture.lean.

Literature events(0)

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Map of approaches(0)

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Attempts(0)

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