Sharp Zero Regions and Hurwitz Criteria for BMV Trace Polynomials

For positive-definite Hermitian matrices \(A,B\), we study the zeros of \(P_m(z)=\operatorname{Tr}((A+zB)^m)\). Loewner bounds \(lB\le A\le uB\) give a sharp dimension-independent zero region: the union of two explicit closed disks. Every point of this region is attained by a commuting two-dimensional pair, while a nonreal boundary zero forces an endpoint block decomposition and commutativity. Optimizing the associated angle gives the optimal universal Hurwitz guarantee \(u/l<\tan^2(\pi/4+\pi/(2m))\) for \(m\ge3\). For arbitrary two-dimensional pairs, including noncommuting ones, a quadratic factorization yields simplicity, an exact stability criterion in terms of \(\operatorname{Tr}(AB)/(\operatorname{Tr} A\operatorname{Tr} B)\), and a count of right-half-plane zeros. We also classify real zeros in every dimension. These are explicit zero-location results motivated by an open-ended AIM problem, not a new proof of the BMV coefficient theorem or a classification for every fixed higher-dimensional pair.