Une combinatoire sous-jacente au théorème des fonctions implicites

AbstractUsing his theory of combinatorial species, André Joyal proved in Advan. in Math. 42 (1981), 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 W. Gröbner (“Die Lie-Reihen und ihre Anwendungen,” D. Verlag d. Wiss., Berlin, 1967; “Contributions to the theory of Lie Series,” Bibliographisches Institut — Mannheim, Hochschultaschenbücher, Mannheim, 1967). 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