A theorem on the escape from the space of hyperbolic polynomials
type: Item
instance of: scholarly article
title: A theorem on the escape from the space of hyperbolic polynomials
zbMATH Open document ID: 0763.58006
DOI: 10.1007/BF02571438
author: Ivan Meguerditchian
published in: Mathematische Zeitschrift
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174451
review text: A hyperbolic polynomial \(P\) is a real polynomial in one variable whose roots are all real. We answer to the following question: For which integers \(s<\deg P\) is it possible to find a polynomial \(Q\) of degree \(s\) such that \(P+Q\) remains hyperbolic? More precisely we will use Arnol'd's terminology: Let \(s\) be an integer such that \(0\leq s<\deg P\), we will say that the monic hyperbolic polynomial \(P\) is globally \(s\)-maximal (respectively globally \(s\)- minimal) if \(P+c_ 0x^ s+\cdots+c_ s\) with \(c_ 0>0\) (respectively \(c_ 0<0)\) is never hyperbolic. The monic hyperbolic polynomial \(P\) is locally \(s\)-maximal (respectively locally \(s\)-minimal) if \(P+c_ 0x^ s+\cdots+c_ s\) with \(c_ 0>0\) (respectively \(c_ 0<0)\) is not hyperbolic for \(c_ 0,\dots,c_ s\) small enough. We prove the equivalence between the local property and the global property and that these properties depend only on the multiplicity vector of \(P\), which we define as follows: If \(P\) is a monic hyperbolic polynomial, then it may be written uniquely as \(P=(X-x_ 1)^{m_ 1}\dots(X-x_ k)^{m_ k}\) with \(x_ 1>x_ 2>\cdots>x_ k\) and the \(k\)-tuple \((m_ 1,\dots,m_ k)\) is called the multiplicity vector of \(P\). To any monic hyperbolic polynomial \(P\) we may associate an integer \(s_ P\) which depends only on the multiplicity-vector of \(P\) as follows: (i) if \(P\) has only simple roots, \(s_ P=-1\). (ii) if not, let \(s_ P=l+\sum_{i/m_ i\geq 2}(m_ i-2)\), where \(l\) is the number of odd sequences of consecutive 1 in the multiplicity vector, considering only sequences which are between two multiplicities greater than or equal to 2. The hyperbolic polynomial \(P\) is called even when its multiplicity vector begins with an even sequence of consecutive 1, and odd otherwise. With these notations we have the following results: Theorem: Let \(P\) be a monic hyperbolic polynomial. a) \(P\) is locally \(s\)-maximal (respectively minimal) if and only if \(P\) is globally \(s\)-maximal…
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 126406
zbMATH Keywords: hyperbolic polynomials
zbMATH Keywords: \(s\)-maximality
MaRDI profile type: Publication
cites work: Q3770650
cites work: Moments of random variables and the equivariant Morse lemma
cites work: On the geometric properties of Vandermonde's mapping and on the problem of moments
OpenAlex ID: W1974761361
description: scientific article; zbMATH DE number 126406