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.
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