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 Panoramas et Synthèses 2011; 152 pp; softcover Number: 33 ISBN-10: 2-85629-346-8 ISBN-13: 978-2-85629-346-1 List Price: US$45 Member Price: US$36 Order Code: PASY/33 This book exposes recent results about hyperbolic polynomials in one real variable, i.e. having all their roots real. It contains a study of the stratification and the geometric properties of the domain in $$\mathbb{R}^n$$ of the values of the coefficients $$a_j$$ for which the polynomial $$P:=x^n+a_1x^{n-1}+\cdots +a_n$$ is hyperbolic. Similar studies are performed w.r.t. very hyperbolic polynomials, i.e. hyperbolic and having hyperbolic primitives of any order, and w.r.t. stably hyperbolic ones, i.e. real polynomials of degree $$n$$ which become hyperbolic after multiplication by $$x^k$$ and addition of a suitable polynomial of degree $$k-1$$. New results are presented concerning the Schur-Szegő composition of polynomials, in particular of hyperbolic ones, and of certain entire functions. The question about the arrangement of the $$n(n+1)/2$$ roots of the polynomials $$P$$, $$P^{(1)}, \ldots, P^{(n-1)}$$ is studied for $$n\leq 5$$ with the help of the discriminant sets $$\mathrm{Res}(P^{(i)},P^{(j)})=0$$. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in hyperbolic polynomials in one variable. Table of Contents Introduction The hyperbolicity domain Very hyperbolic and stably hyperbolic polynomials The Schur-Szegö composition and the mapping $$\Phi$$ Root arrangements and the Rolle theorem Testing hyperbolicity Bibliography