Mathematics – Classical Analysis and ODEs
Scientific paper
2003-12-17
Rocky Mountain J. Math. 38 (2008), 809-834
Mathematics
Classical Analysis and ODEs
final version, to appear in Rocky Mountain J. Math.; 14 pages, no figures, LaTeX2e
Scientific paper
The strong spectral order induces a natural partial ordering on the manifold $H_{n}$ of monic hyperbolic polynomials of degree $n$. We prove that twisted root maps associated with linear operators acting on $H_{n}$ are G\aa rding convex on every polynomial pencil and we characterize the class of polynomial pencils of logarithmic derivative type by means of the strong spectral order. Let $A'$ be the monoid of linear operators that preserve hyperbolicity as well as root sums. We show that any polynomial in $H_{n}$ is the global minimum of its $A'$-orbit and we conjecture a similar result for complex polynomials.
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