Continuity of the roots of a nonmonic polynomial and applications in multivariate stability theory

TL;DR

This paper investigates how the roots of nonmonic polynomials change continuously with their coefficients, providing elementary proofs and applications to multivariate stability theory, relevant in physics, engineering, and mathematics.

Contribution

It introduces elementary methods to analyze root continuity of nonmonic polynomials, including cases where polynomial degree varies, with applications to stability and eigenvalue problems.

Findings

Established qualitative and quantitative root continuity results

Applied findings to multivariate stability theory and hyperbolic polynomials

Extended analysis to cases with changing polynomial degree

Abstract

We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach gives both qualitative and quantitative results in the case that the degree of the unperturbed polynomial can change under a perturbation of its coefficients, a case that naturally occurs, for instance, in stability theory of polynomials, singular perturbation theory, or in the perturbation theory for generalized eigenvalue problems. An application of our results in multivariate stability theory is provided which is important in, for example, the study of hyperbolic polynomials or realizability and synthesis problems in passive electrical network theory, and will be of general interest to mathematicians as well as physicists and engineers.