A Symmetric Counterexample to Brändén's Majorization Conjectures for Mixed Characteristic Polynomials

In an extended abstract in Oberwolfach Rep. 14 (2017), no. 1, 771–862 (pp. 825–826), Brändén considered the set Dm,d of stable homogeneous polynomials P of degree d in m variables with ∂P/∂x_i(1) = d/m for all i, and the zero vector λ(P) of the mixed characteristic polynomial χ_P(t) = ∏_i (1 − ∂/∂x_i)P evaluated at x = t·1. He conjectured that λ(P) is majorized by λ(P_∞), where P_∞ = (x_1 + ⋯ + x_m)^d/m^d (Conjecture 2), and, for symmetric P, that λ(Ψ_r P) is majorized by λ(Ψ_s P) for 0 ≤ r ≤ s along a flow Ψ_s tending to P_∞ (Conjecture 5). We show that both conjectures fail for the symmetric polynomial P = (x_1 + x_2)(x_1 + x_3)(x_2 + x_3)/8 in D3,3, by exact proofs that need no numerical computation. Here χ_P = (t − 1)(4t² − 8t + 1)/4, and λ_1(P) + λ_2(P) = 2 + √3/2 exceeds the same sum for P_∞ because 9√3 < 16. Along the flow, Ψ_s P is again a product of three linear forms, χΨ_s P is an explicit cubic with positive discriminant, and λ_1 + λ_2 of Ψ_s P is smaller than that of P for every s > 0; so Conjecture 5 fails with r = 0 and every s > 0. The flow is printed as exp(s(I − T)), a sign under which the source's Lemma 3 fails; we prove that lemma for exp(s(T − I)), and show that Conjecture 5 fails under either sign. Conjecture 2 itself is not new: it was reportedly refuted in a 2017 KTH degree project (Samuel Zackrisson, DiVA diva2:1106332; full text not read), according to its abstract by way of Wanless's counterexamples to the Holens–Đoković permanent conjecture, and according to a search-engine excerpt also by an explicit example. Our example satisfies the Holens–Đoković inequalities, so it does not arise from Wanless's counterexamples; we claim no priority for it. We found no refutation of Conjecture 5 in the sources searched; since the thesis could not be read, we cannot exclude that it also treats Conjecture 5. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-15219-013 (Brändén, The van der Waerden and Kadison-Singer conjectures with symmetries, Oberwolfach Rep. 14 (2017), no. 1, 771–862, pp. 825–826).