We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde, and other structured matrices, Stietjes-and Jacobitype continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.Introduction. This paper offers a glimpse at several classical topics going back to Descartes, Gauss, Stieltjes, Hermite, Hurwitz, and Sylvester (see, e.g., [19,80,28,81,82]), all connected by the idea that behavior of polynomials can be analyzed via algebraic constructs involving their coefficients. Thus a number of interrelated algebraic constructs were built, including Hurwitz, Toeplitz, and Hankel matrices, the corresponding quadratic forms, and the corresponding continued fractions. The linear-algebraic properties of these objects were shown to be intimately related to root localization of polynomials such as stability, whereby the zeros of a polynomial avoid a specific half-plane, or hyperbolicity, whereby the zeros lie on a specific line. These methods gave rise to several well-known tests of root localization, such as the Routh-Hurwitz algorithm or the Lienard-Chipart test.In the 20th century, this line of research was developed further by Schur [24], Pólya [78], Krein [59,60,34], and others (see, e.g., [43]), leading to important notions of total positivity, Pólya frequency sequences, stability preservers, etc. A very important part of that research effort was devoted to entire functions, in particular, the

We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only real and simple zeroes. All proofs are given using properties of rational functions mapping the upper half-plane of the complex plane to the lower half-plane. Matrices with self-interlacing spectra and other applications of generalized Hurwitz polynomials are discussed.

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.Comment: 10 page

For a given real entire function ϕ with finitely many nonreal zeroes in the class U * 2n , n 0, we establish a connection between the number of real zeroes of the functions

We prove that for any circulant matrix C of size n × n with the monic characteristic polynomial p(z), the spectrum of its (n − 1) × (n − 1) submatrix Cn−1 constructed with first n − 1 rows and columns of C consists of all critical points of p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R. Pereira and S. Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S.