Sign patterns that require HnHn exist for each n ≥ 4

The refined inertia of a square real matrix A   is the ordered 4-tuple (n+,n−,nz,2np)(n+,n−,nz,2np), where n+n+ (resp., n−n−) is the number of eigenvalues of A   with positive (resp., negative) real part, nznz is the number of zero eigenvalues of A  , and 2np2np is the number of nonzero pure imaginary eigenvalues of A  . For n≥3n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)}Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. We say that an n×nn×n sign pattern AA requires HnHn if Hn={ri(B)|B∈Q(A)}Hn={ri(B)|B∈Q(A)}. Bodine et al. conjectured that no n×nn×n irreducible sign pattern that requires HnHn exists for n   sufficiently large, possibly n≥8n≥8. However, for each n≥4n≥4, we identify three n×nn×n irreducible sign patterns that require HnHn, which resolves this conjecture.