A continuity proof of Rudin's theorem for polynomials and a generalization

Abstract

We assign to each nonzero complex polynomial the minimum of the absolute values of its roots. We show the simple principle that this minimum depends continuously on the coefficients of the polynomial and is sufficiently powerful to give a very elementary proof of Rudin's stability theorem for multivariable polynomials. Moreover, we show that the polynomial version of a generalization on Rudin's theorem due to Hertz and Zeheb is obtained as a consequence of this principle.