An asymptotic property of the roots of polynomials

Abstract

It is shown that if the imaginary parts of the roots λ j ( s ) {\lambda _j}(s) of a polynomial P ( λ , s ) , s ∈ R n P(\lambda ,s),s \in {R^n} , are unbounded for large | s | |s| , then they are in fact unbounded along a one-parameter algebraic curve s = s ( R ) 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.