Complex-Analytic Theory of the m-Function*

In this paper, we consider the determinant of the multivariable return difference Nyquist map, crucial in defining the complex m-function, as a holomorphic function defined on a polydisk of uncertainty. The key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping. From this single result only, we show that, in the diagonal perturbation case, all preimage points of the boundary of the Horowitz template are included in the distinguished boundary of the polydisk. In the block-diagonal perturbation case, where each block is norm-bounded by one, a preimage of the boundary is shown to be a unitary matrix in each block. Finally, some algebraic geometry, together with the Weierstrass preparation theorem, allows us to show that the deformation of the crossover under holomorphic.variations of ‘‘certain’’ parameters is continuous.