Multivariate Padé-Bergman approximants

Summary. We define the multivariate Padé-Bergman approximants (also called Padé approximants) and prove a natural generalization of de Montessus de Ballore theorem. Numerous definitions of multivariate Padé approximants have already been introduced. Unfortunately, they all failed to generalize de Montessus de Ballore theorem: either spurious singularities appeared (like the homogeneous Padé [3,4], or no general convergence can be obtained due to the lack of consistency (like the equation lattice Padé type [3]). Recently a new definition based on a least squares approach shows its ability to obtain the desired convergence [6]. We improve this initial work in two directions. First, we propose to use Bergman spaces on polydiscs as a natural framework for stating the least squares problem. This simplifies some proofs and leads us to the multivariate Padé approximants. Second, we pay a great attention to the zero-set of multivariate polynomials in order to find weaker (although natural) hypothesis on the class of functions within the scope of our convergence theorem. For that, we use classical tools from both algebraic geometry (Nullstellensatz) and complex analysis (analytic sets, germs). Received December 4, 2001 / Revised version received January 2, 2002 / Published online April 17, 2002