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)=Tr((A+zB)^m). Loewner bounds lB<=A<=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 =3. For arbitrary two-dimensional pairs, including noncommuting ones, a quadratic factorization yields simplicity, an exact stability criterion in terms of Tr(AB)/(Tr A 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 AIM-ANALYSIS-0138 (AIM Matrix Theory, Problem 2.1), not a new proof of the BMV coefficient theorem or a classification for every fixed higher-dimensional pair. The universal locus ranges over pairs. Classical methods and the inherited commuting example are credited; folklore and unlocated prior art are not excluded. Seven-page English preprint with source and reproducibility files, including 1580 exact finite diagnostics. AI assistance was used in derivation, literature search, checking and exposition. Originating-researcher self-audit only: this preprint is unrefereed, not formally verified, and makes no absolute-priority claim.