Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations (Q1113299)

Let G be the complexification of a connected compact n-dimensional Lie group K and let \(\pi_ 1,...,\pi_ n\) for finite-dimensional holomorphic representations of G. The system \(f_ 1=...=f_ n=0\) is considered, where \(f_ i\) is a matrix-valued function of the representation \(\pi_ j\). All the systems that lie outside a certain algebraic hypersurface in the space of such systems have the same number of roots \(N(\pi_ 1,...,\pi_ n)\). The author deduces a geometrical formula for this number. The particular case \(G=(C\setminus 0)^ n\) is considered more closely.