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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Location of the zeros of a polynomial relative to certain disks


Author: R. C. Riddell
Journal: Trans. Amer. Math. Soc. 205 (1975), 37-45
MSC: Primary 30A08
DOI: https://doi.org/10.1090/S0002-9947-1975-0364603-5
MathSciNet review: 0364603
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Abstract: The zeros of the complex polynomial $ P(z) = {z^n} + \Sigma {\alpha _i}{z^{n - 1}}$ are studied under the assumption that some $ \vert{\alpha _k}\vert$ is large in comparison with the other $ \vert{\alpha _i}\vert$. It is shown under certain conditions that $ P(z)$ has $ n - k$ zeros in $ \vert z\vert \leq {m_ - }$ and $ k$ zeros in $ \vert z\vert \geq {m_ + }$, where $ {m_ - } < {m_ + } \leq \vert{\alpha _k}{\vert^{1/k}}$; and under suitably strengthened conditions, one of the $ k$ zeros of larger modulus is shown to lie in each of the $ k$ disks $ \vert z - {( - {\alpha _k})^{1/k}}\vert \leq R$, where $ {m_ - } + R < \vert{\alpha _k}{\vert^{1/k}}$.


References [Enhancements On Off] (What's this?)

  • [1] J. Dieudonné, La théorie analytique des polynomes d'une variable, Mémor. Sci. Math. 93 (1938), 1-71.
  • [2] Maurice Parodi, La localisation des valeurs caractéristiques des matrices et ses applications. Préface de H. Villat, Traité de Physique Théorique et de Physique Mathématique, XII, Gauthier-Villars, Paris, 1959 (French). MR 0110719
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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0364603-5
Keywords: Dominant coefficient, zero-free annulus, disk which isolates a single zero
Article copyright: © Copyright 1975 American Mathematical Society