Geodesic
On Descartes' rule of signs for hyperbolic polynomials
Serdica Mathematical Journal, Tome 49 (2023) no. 4, pp. 251-268.

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We consider univariate real polynomials with all real roots and with two sign changes in the sequence of their coefficients which are all non-vanishing. Assume that one of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree \(d\) polynomials have 2 positive and \(d-2\) negative roots. We consider the possible sequences of the moduli of their roots on the real positive half-axis. When these moduli are distinct, we give the exhaustive answer to the question which positions can the moduli of the two positive roots occupy.
Keywords: real polynomial in one variable, hyperbolic polynomial, sign pattern, Descartes' rule of signs, 26C10 
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     title = {On {Descartes'} rule of signs for hyperbolic polynomials},
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Kostov, Vladimir. On Descartes' rule of signs for hyperbolic polynomials. Serdica Mathematical Journal, Tome 49 (2023) no. 4, pp. 251-268. http://geodesic.mathdoc.fr/item/SMJ2_2023_49_4_a2/