Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An asymptotic property of the roots of polynomials

Author: Hermann Flaschka
Journal: Proc. Amer. Math. Soc. 27 (1971), 451-456
MSC: Primary 35L40; Secondary 30A08
MathSciNet review: 0303102
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Abstract: It is shown that if the imaginary parts of the roots $ {\lambda _j}(s)$ of a polynomial $ P(\lambda ,s),s \in {R^n}$, are unbounded for large $ \vert s\vert$, then they are in fact unbounded along a one-parameter algebraic curve $ s = s(R)$. The result may be used to reduce certain questions about polynomials in several variables to an essentially one-dimensional form; this is illustrated by an application to hyperbolic polynomials.

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

  • [1] Robert J. Walker, Algebraic curves, Dover Publications, Inc., New York, 1962. MR 0144897
  • [2] Gilbert Strang, On multiple characteristics and the Levi-Lax conditions for hyperbolicity, Arch. Rational Mech. Anal. 33 (1969), 358–373. MR 0243185
  • [3] Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
  • [4] L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York and Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • [5] S. Leif Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat. 8 (1969), 145–162. MR 0271538
  • [6] Anneli Lax, On Cauchy’s problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math. 9 (1956), 135–169. MR 0081406
  • [7] Masaya Yamaguti, Le problème de Cauchy et les opérateurs d’intégrale singulière, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 32 (1959), 121–151 (French). MR 0109259

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Keywords: Roots of polynomials, Seidenberg-Tarski theorem, hyperbolic polynomials
Article copyright: © Copyright 1971 American Mathematical Society