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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.

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

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