Cavers, Michael Polynomial stability and potentially stable patterns. (English) Zbl 1458.15055 Linear Algebra Appl. 613, 87-114 (2021). Summary: A polynomial (resp. matrix) is stable if all of its roots (resp. eigenvalues) have negative real parts. A sign (resp. nonzero) pattern \(\mathcal{A}\) is a matrix with entries in \(\{+,-,0\}\) (resp. \(\{\ast,0\})\). If there exists a real stable matrix with pattern \(\mathcal{A}\), then \(\mathcal{A}\) is potentially stable. This paper first shows that if \(p(t)=c_0t^n+c_1t^{n-1}+\cdots+c_n\) is a (real) stable polynomial with \(c_0>0\), then \(c_ic_j>c_kc_\ell\) for every \(0\leq k<i\leq j\leq n\) such that \(ij+k\ell\) is even and \(\ell=i+j-k\leq n\). 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 \(m_n\), the minimum number of nonzero entries in an irreducible potentially stable pattern of order \(n\). It is shown that \(m_8=12\) and conjectured that \(m_n\leq\lceil 3n/2\rceil\) for \(n\geq 2\). To support this conjecture, a family of irreducible patterns with exactly \(\lceil 3n/2\rceil\) 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. Cited in 2 Documents MSC: 15B35 Sign pattern matrices 15A18 Eigenvalues, singular values, and eigenvectors 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C20 Directed graphs (digraphs), tournaments Keywords:polynomial stability; nonzero pattern; sign pattern; potentially stable; signed digraph PDFBibTeX XMLCite \textit{M. Cavers}, Linear Algebra Appl. 613, 87--114 (2021; Zbl 1458.15055) Full Text: DOI References: [1] Berliner, A. H.; Olesky, D. 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