We prove that a hyperbolic monic polynomial whose coefficients are functions of class $C^r$ of a parameter $t$ admits roots of class $C^1$ in $t$, if $r$ is the maximal multiplicity of the roots as $t$ varies. Moreover, if the coefficients are functions of \(t\) of class $C^{2r}$, then the roots may be chosen two times differentiable at every point in $t$. This improves, among others, previous results of Bron\v ste\u \i n, Mandai, Wakabayashi and Kriegl, Losik and Michor.
On the regularity of the roots of hyperbolic polynomials
COLOMBINI, FERRUCCIO;
2012-01-01
Abstract
We prove that a hyperbolic monic polynomial whose coefficients are functions of class $C^r$ of a parameter $t$ admits roots of class $C^1$ in $t$, if $r$ is the maximal multiplicity of the roots as $t$ varies. Moreover, if the coefficients are functions of \(t\) of class $C^{2r}$, then the roots may be chosen two times differentiable at every point in $t$. This improves, among others, previous results of Bron\v ste\u \i n, Mandai, Wakabayashi and Kriegl, Losik and Michor.File in questo prodotto:
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