Le grand théorème de Picard pour les multifonctions analytiques finies
Volume 77 / 2001
Abstract
Let be a domain of the complex plane containing the origin. The famous great theorem of Émile Picard asserts that if h is holomorphic on D\setminus\{ 0\} , with an essential singularity at 0, then the image under h of any pointed neighbourhood of 0 covers all the complex plane, with at most one exception. Introducing the concept of essential singularity for analytic multifunctions, we extend this theorem to a finite analytic multifunction K, of degree N, defined on D\setminus\{ 0\}. In this case \bigcup _{0 < |\lambda |< r}K (\lambda ) covers all the complex plane, with at most 2N-1 exceptions. In particular, this theorem can be used in the case of N\times N matrices whose entries are holomorphic on D\setminus \{ 0\} with essential singularities at 0. In this case, if their spectra avoid 2N points on a pointed neighbourhood of 0, these spectra must be constant.
