Topics on hyperbolic polynomials in one variable (Q2900756)

A real polynomial in one variable is said to be hyperbolic if it has only real roots. Such polynomials arise in a natural way in many contexts: familiar examples include characteristic polynomials of symmetric matrices, as well as classic orthogonal polynomials. The goal of the present book is to expose a number of recent results on hyperbolic polynomials. Only a part of the proofs of these results is presented, the reader being referred to the references for complete proofs.NEWLINENEWLINEA first part of the book contains a study of the stratification and geometric properties of the domain of coefficients \((a_1,\ldots,a_n)\) in \(\mathbb R^n\) for which the polynomial \( x^n+a_1x^{n-1}+\cdots+a_n\) is hyperbolic. Similar studies are performed for so-called very hyperbolic polynomials, that is, hyperbolic polynomials having hyperbolic primitives of any order, as well as for stably hyperbolic polynomials, that is, real polynomials which become hyperbolic after multiplication by the monomial \(x^k\) and addition of a suitable polynomial of degree \(k-1\), for some integer \(k\). New results are presented concerning the Schur-Szegö composition of polynomials, in particular of hyperbolic ones, and of certain entire functions. Finally, the possible arrangements of the \({1\over 2}n(n+1)\) roots of the polynomials \( P, P', \ldots, P^{(n-1)}\) are studied for \( n\leq 5 \), with the help of the discriminant sets \( \text{Res} (P^{(i)},P^{(j)})=0 \).