Complex-analytic theory of the mu-function

In this paper, we consider the determinant of the multivariable return diff erence Nyquist map. crucial in defining the complex mu-function, as a holom orphic function defined on a polydisk of uncertainty. The key property of h olomorphic functions of several complex variables that is crucial in our ar gument 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 bounda ry of the Horowitz template are included in the distinguished boundary of t he polydisk. In the block-diagonal perturbation case. where each block is n orm-bounded by one, a preimage of the boundary is shown to be a unitary mat rix in each block. Finally, some algebraic geometry, together with the Weie rstrass preparation theorem, allows us to show that the deformation of I-he crossover under (holomorphic) variations of "certain" parameters is contin uous. (C) 1999 Academic Press.