# Seminar in Real & Complex Geometry

## A strong version of implicit function theorem

Abstract

 Consider the system of equations F(x,y)=0. The classical Implicit Function Theorem starts from the assumption: "the derivative F'_y(x,0) is nondegenerate/right-invertible". This condition is far from necessary. It has been weakened by J.C.Tougeron (Tougeron's implicit function theorem) and then further by B.Fischer. Another direction was to extend the theorem to a more general class of rings, beyond the rings of analytic/smooth functions. We obtain the "weakest possible" condition that ensures the solvability of F(x,y)=0. The condition is in terms of some suitable filtration, it is necessary and sufficient for the existence of "good" (differentiable) solutions. In the simplest cases this filtration condition reproduces Tougeron's/Fisher's theorems. For more delicate filtrations we get significant strengthenings. This gives a version of the strong Implicit Function Theorem in the category of filtered groups. In particular, we get the solvability for a broad class of (commutative, associative) rings. Finally, we prove the Artin-type approximation theorem: if a system of C^\infty equations has a formal solution and the derivative F'_y(x,0) satisfies a Lojasiewicz-type condition then the system has a C^\infty-solution. Joint work with G. Belitski.