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We study continuity of the roots of nonmonic polynomials as a function oftheir coefficients using only the most elementary results from an introductorycourse in real analysis and the theory of single variable polynomials. Ourapproach gives both qualitative and quantitative results in the case that thedegree of the unperturbed polynomial can change under a perturbation of itscoefficients, a case that naturally occurs, for instance, in stability theoryof polynomials, singular perturbation theory, or in the perturbation theory forgeneralized eigenvalue problems. An application of our results in multivariatestability theory is provided which is important in, for example, the study ofhyperbolic polynomials or realizability and synthesis problems in passiveelectrical network theory, and will be of general interest to mathematicians aswell as physicists and engineers.

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