Joint continuity of division of smooth functions. I: Uniform lojasiewicz estimates

In this paper we study the question of the existence of a continuous inverse to the multiplication mapping (formula presented) defined on pairs of C00 functions on a manifold M. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of (formula presented) mapping of the manifold M into an analytic manifold N and p is a fixed analytic function on N. We prove a theorem adequate for application to the study of inverting the mapping (formula presented), where A is a vector valued C function and A is a square matrix valued C00 function on M whose determinant may vanish on a nowhere dense set.