Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1971-0303102-5
MathSciNet review: 0303102
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] R. J. Walker, Algebraic curves, Dover, New York, 1962. MR 26 #2438. MR 0144897 (26:2438)
  • [2] G. Strang, On multiple characteristics and the Levi-Lax conditions for hyperbolicity, Arch. Rational Mech. Anal. 33 (1969), 358-373. MR 39 #4509. MR 0243185 (39:4509)
  • [3] A. Friedman, Generalized functions and partial differential equations, Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 29 #2672. MR 0165388 (29:2672)
  • [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] L. Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat. 8 (1969), 145-162. MR 0271538 (42:6421)
  • [6] A. Lax, On Cauchy's problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math 9 (1956), 135-169. MR 18, 397. MR 0081406 (18:397b)
  • [7] M. Yamaguti, Le problème de Cauchy et les opérateurs d'intégrale singulière, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1959), 121-151. MR 22 #146. MR 0109259 (22:146)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35L40, 30A08

Retrieve articles in all journals with MSC: 35L40, 30A08


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0303102-5
Keywords: Roots of polynomials, Seidenberg-Tarski theorem, hyperbolic polynomials
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society