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

We assign to each nonzero complex polynomial the minimum of the absolute va lues of its roots. We show the simple principle that this minimum depends c ontinuously on the coefficients of the polynomial is sufficiently powerful to give a very elementary proof of Rudin's stability theorem for multivaria ble polynomials. Moreover, we show that the polynomial version of a general ization of Rudin's theorem due to Hertz and Zeheb is obtained as a conseque nce of this principle.