Hyperbolic polynomials and spectral order

Julius Borcea; Boris Shapiro

doi:10.1016/j.crma.2003.10.007

Mathematical Analysis

Hyperbolic polynomials and spectral order

Presented by: Jean-Pierre Kahane

Julius Borcea ¹ ; Boris Shapiro ¹

¹ Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 693-698

Abstract (Original language)
Résumé

The spectral order on $ℝ$ induces a partial ordering on the manifold $ℋ_{n}$ of monic hyperbolic polynomials of degree n. We show that the semigroup $\tilde{𝒮}$ generated by differential operators of the form $(1 - λ \frac{d}{d x}) e^{λ d / d x}$ , $λ \in ℝ$ , acts on the poset $ℋ_{n}$ in an order-preserving fashion. We also show that polynomials in $ℋ_{n}$ are global minima of their respective $\tilde{𝒮}$ -orbits and we conjecture that a similar result holds even for complex polynomials. Finally, we show that only those pencils of polynomials in $ℋ_{n}$ which are of logarithmic derivative type satisfy a certain local minimum property for the spectral order.

L'ordre spectral sur $ℝ$ induit un ordre partiel sur la variété $ℋ_{n}$ des polynômes hyperboliques de degré n dont le coefficient dominant est égal à un. On montre que cet ordre est préservé par l'action sur $ℋ_{n}$ du semigroupe $\tilde{𝒮}$ engendré par les opérateurs différentiels du type $(1 - λ \frac{d}{d x}) e^{λ d / d x}$ , $λ \in ℝ .$ On démontre aussi que tout polynôme de $ℋ_{n}$ est le minimum global de son $\tilde{𝒮}$ -orbite et on propose une conjecture selon laquelle un résultat similaire serait valable dans le cas des polynômes à coefficients complexes. On montre enfin que de tous les faisceaux de polynômes dans $ℋ_{n}$ , seulement ceux qui sont associés aux dérivées logarithmiques satisfont une certaine propriété de minimum local pour l'ordre spectral.

Received: 2003-03-11
Accepted: 2003-10-07
Published online: 2003-11-07

DOI: 10.1016/j.crma.2003.10.007

Author's affiliations:

Julius Borcea ¹ ; Boris Shapiro ¹

¹ Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

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Julius Borcea; Boris Shapiro. Hyperbolic polynomials and spectral order. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 693-698. doi: 10.1016/j.crma.2003.10.007

References
Cited by

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[3] A.W. Marshall; I. Olkin Inequalities: Theory of Majorization and Its Applications, Math. Sci. Engrg., 143, Academic Press, New York, 1979

[4] N. Obreschkoff Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, 1963

[5] S. Sherman On a theorem of Hardy, Littlewood, Pólya, and Blackwell, Proc. Nat. Acad. Sci. USA, Volume 37 (1951), pp. 826-831

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