**Panoramas et Syntheses**

Volume: 33;
2011;
152 pp;
Softcover

MSC: Primary 12; 26; 30;
**Print ISBN: 978-2-85629-346-1
Product Code: PASY/33**

List Price: $45.00

AMS Member Price: $36.00

# Topics on Hyperbolic Polynomials in One Variable

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*Vladimir Petrov Kostov*

A publication of the Société Mathématique de France

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.