An Asymptotic Property of the Roots of Polynomials

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 $|s|$, 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.