Une combinatoire sous-jacente au théorème des fonctions implicites

Using his theory of combinatorial species, 3.], 1–82 a combinatorial form of the classical multidimensional implicit function theorem. His theorem asserts the existence and (strong) unicity of species satisgying systems of combinatorial equations of a very general type. We present an explicit construction of these species by using a suitable combinatorial version of the Lie Series in the sense of 1. and 2.]. The approach constitutes a generalization of the method of “éclosions” (bloomings) which was used by the author in (J. Combin. Theory Ser. A 39, No. 1 (1985), 52–82), to study multidimensional power series reversion. Remarks concerning the applicability of the method to solve certain combinatorial differential equations are also made at the end of the work.