# Topics on Hyperbolic Polynomials in One Variable

Abstract : 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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Books

https://hal.archives-ouvertes.fr/hal-00866240
Submitted on : Thursday, September 26, 2013 - 1:20:02 PM
Last modification on : Monday, October 12, 2020 - 2:28:05 PM

### Identifiers

• HAL Id : hal-00866240, version 1

### Citation

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

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