Geodesic
Hyperbolic polynomials and canonical sign patterns
Serdica Mathematical Journal, Tome 46 (2020) no. 2, pp. 135-150.

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A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly \(c\) positive and exactly \(p\) negative roots counted with multiplicity, where \(c\) and \(p\) are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all \(c+p\) roots are distinct and ordered on the positive half-axis, then at which positions can the \(p\) moduli of negative roots be depending on the positions of the positive and negative signs of the coefficients of the polynomial? We are especially interested in the choices of these signs for which exactly one order of the moduli of the roots is possible.
Keywords: real polynomial in one variable, hyperbolic polynomial, sign pattern, Descartes' rule of signs, 26C10, 30C15 
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     author = {Kostov, Vladimir},
     title = {Hyperbolic polynomials and canonical sign patterns},
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Kostov, Vladimir. Hyperbolic polynomials and canonical sign patterns. Serdica Mathematical Journal, Tome 46 (2020) no. 2, pp. 135-150. http://geodesic.mathdoc.fr/item/SMJ2_2020_46_2_a3/