Open Sets of Linear Systems with n = mp That Are Not Stabilizable by Static Output Feedback: New Cases of a Problem of Eremenko

A real linear system x' = Ax + Bu, y = Cx with n states, p inputs and m outputs is stabilizable by static output feedback if, for some real matrix K, all eigenvalues of A + BKC have negative real part. Generic systems are stabilizable when n < mp, and when n = mp and the degree d(m,p) of the Grassmannian is odd. Eremenko asks for which (m,p) the generic system with n = mp is stabilizable; the only pair known to fail was (2,2) (Molander; Byrnes and Anderson). We settle some new cases, all negatively; the classification remains open. For (m,p) = (2,4), (2,5) and (3,3) we give explicit integer polynomial matrices with certificates, checkable in exact arithmetic, which show by the criterion of Byrnes and Anderson that the systems that are not stabilizable contain a non-empty open set; the same follows for (4,2) and (5,2). We prove the same for (2,p), (p,2), (4,p) and (p,4) with every even p, which answers a question of Eremenko and Gabrielov when min(m,p) ∈ {2,4}; here the examples are complex polynomial curves regarded as real ones, and the proof is a count of dimensions. As a by-product, real pole placement is not generically possible for (2,5), (5,2) and (3,3), which are cases of a conjecture of Rosenthal and Sottile. We also note that the 2 × 4 matrix printed in the problem file dated October 3, 2026 differs in a sign from the example of Byrnes and Anderson and is not a counterexample. This is an unrefereed note.