Polynomial stability and potentially stable patterns

Abstract

A polynomial (resp. matrix) is stable if all of its roots (resp. eigenvalues) have negative real parts. A sign (resp. nonzero) pattern A is a matrix with entries in {+,−,0} (resp. {⁎,0}). If there exists a real stable matrix with pattern A, then A is potentially stable. This paper first shows that if p(t)=c0tn+c1tn−1+⋯+cn is a (real) stable polynomial with c0>0, then cicj>ckcℓ for every 0≤k<i≤j≤n such that ij+kℓ is even and ℓ=i+j−k≤n. Using this, certain patterns are shown to not be potentially stable based solely on the cyclic structure of their digraphs. Next, bounds are given on mn, the minimum number of nonzero entries in an irreducible potentially stable pattern of order n. It is shown that m8=12 and conjectured that mn≤⌈3n/2⌉ for n≥2. To support this conjecture, a family of irreducible patterns with exactly ⌈3n/2⌉ nonzero entries is described and demonstrated to be potentially stable for small values of n. Finally, the potentially stable nonzero patterns of order at most 4 are characterized.