bibkey: kovacec2026strongermoc
authors: A. Kovačec
year: 2026
title: A conjecture more precise and stronger than the one by Marcus and de Oliveira
doi: null
url: https://www.mat.uc.pt/preprints/ps/p2611.pdf
claim: "Conjecture 7 asserts ∆(A, B) ⊆ closure(J(A, B)) whenever the exterior boundary of the σ-point graph is a Jordan curve."
strata_touched:
  - D5/S3/Quantum/Matrix/KovacecStrengthenedMarcusDeOliveiraRefutation
license: citation-only
triage: anchor

A conjecture more precise and stronger than the one by Marcus and de Oliveira

Verified locator

Source: https://www.mat.uc.pt/preprints/ps/p2611.pdf

DMUC Preprint 26-11, v1. Conjecture 7 and Definitions 6b are on p. 5; the σ-points and determinant range are on p. 2; the transposition graph and Definitions 6a are on p. 4.

Source statements

Page 2, §1:

Often one writes z_σ = z_σ(A, B) = ∏ᵢ₌₁ⁿ (aᵢ + b_{σ(i)}), and calls these points σ-points.

Putting ∆ = ∆(A, B) = {det(D_A + U D_B U*) : U ∈ U(n)}, we can say this also this way: Conjecture 1’’. For any n-multisets of complex numbers A, B we have ∆(A, B) ⊆ C(A, B).

Page 4, §2:

The graph joins permutations “that differ by exactly one transposition” by the closed segments between their σ-points.

A metric (linear) graph G = G(V, E) is a subset of the plane consisting of a family V of complex numbers (or pairs of reals) as vertices and a family E of segments [v, w] with v, w ∈ V as edges.

Page 5, Definitions 6b:

A point p is an exterior point of a finite metric linear graph G if for all M > 0 there is a point q at distance M from p such that [p, q] does not intersect any edge of G. The set of all exterior points is the exterior of G.

Page 5, Conjecture 7 (strengthened MOC):

Given any two n-multisets of complex numbers A, B such that a Jordan curve J and the bounded component J(A, B) are defined, there holds ∆(A, B) ⊆ closure(J(A, B)).

The graph is the union of closed segments joining σ-points whose permutations differ by one transposition. A Jordan curve is represented as the range of a continuous injective map from Circle to ℂ. The inclusion is expressed for every bounded connected component of the complement of the exterior frontier. The complex multisets are represented by functions Fin n → ℂ, retaining multiplicities.

The refutation concerns Conjecture 7. The original Marcus–de Oliveira convex-hull conjecture is a separate statement.