Hyperbolic polynomials and spectral order

Abstract

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 S generated by differential operators of the form (1−λ d dx ) e λ d/ dx , λ∈ 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 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. To cite this article: J. Borcea, B. Shapiro, C. R. Acad. Sci. Paris, Ser. I 337 (2003).