Abstract :
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.
