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 \(\mathcal{D}_{m,d}\) of stable homogeneous polynomials \(P\) of degree \(d\) in \(m\) variables with \(\partial P\)\(/\partial x_i(\mathbf{1})\)\({}=d\)\(/m\) for all \(i\), and the zero vector \(\lambda(P)\) of the mixed characteristic polynomial \(\chi_P(t)\)\({}=\prod_i(1\)\({}-\partial\)\(/\partial x_i)P|_{x=t\mathbf{1}}\). He conjectured that \(\lambda(P)\) is majorized by \(\lambda(P_\infty)\), where \(P_\infty\)\({}=(x_1\)\({}+\)\(\dots\)\({}+x_m)^d\)\(/m^d\) (Conjecture 2), and, for symmetric \(P\), that \(\lambda(\Psi_rP)\) is majorized by \(\lambda(\Psi_sP)\) for \(0\le r\le s\) along a flow \(\Psi_s\) tending to \(P_\infty\) (Conjecture 5). We show that both conjectures fail for the symmetric polynomial \(P\)\({}=(x_1\)\({}+x_2)(x_1\)\({}+x_3)(x_2\)\({}+x_3)\)\(/8\)\({}\in\mathcal{D}_{3,3}\), by exact proofs that need no numerical computation. Here \(\chi_P\)\({}=(t\)\({}-1)(4t^2\)\({}-8t\)\({}+1)\)\(/4\), and \(\lambda_1(P)\)\({}+\lambda_2(P)\)\({}=2\)\({}+\sqrt3\)\(/2\) exceeds the same sum for \(P_\infty\) because \(9\sqrt3<16\). Along the flow, \(\Psi_sP\) is again a product of three linear forms, \(\chi_{\Psi_sP}\) is an explicit cubic with positive discriminant, and \(\lambda_1+\lambda_2\) of \(\Psi_sP\) 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.