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Ouvrages Année : 2011

Topics on Hyperbolic Polynomials in One Variable

Vladimir Kostov

Résumé

The 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-Szeg\H{o} composition of polynomials, in particular of hyperbolic ones, and of certain entire functions. The question what can be 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 $Res(P^{(i)},P^{(j)})=0$.
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Dates et versions

hal-00866240 , version 1 (26-09-2013)

Identifiants

  • HAL Id : hal-00866240 , version 1

Citer

Vladimir Kostov. Topics on Hyperbolic Polynomials in One Variable. SMF, pp.1-141, 2011. ⟨hal-00866240⟩

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