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We study the {\it regular} action of an analytic pseudo-group oftransformations on the space of germs of various analytic objects of localanalysis and local differential geometry. We fix a homogeneous object $F_0$ andwe are interested in an analytic normal form for the whole affine space $\{F_0+ h.o.t.\}$. We prove that if the cohomological operator defined by $F_0$ hasthe big denominators property and if a formal normal form is well chosen thenthis formal normal form holds in analytic category. We also define bigdenominators in systems of nonlinear PDEs and prove a theorem on local analyticsolvability of systems of nonlinear PDEs with big denominators. %This theorem,which a priori has nothing to do with local classification problems,generalizesthe theorem on analytic normal forms. Moreover, we prove that if thedenominators grow "relatively fast", but not fast enough to satisfy the bigdenominator property, then we have a normal form, respectively localsolvability of PDEs, in a formal Gevrey category. We illustrate our theorems byexplanation of known results and by new results in the problems of localclassification of singularities of vector fields, non-isolated singularities offunctions, tuples of germs of vector fields, local Riemannian metrics andconformal structures.

Big denominators and analytic normal forms with an appendix by M. Zhitomirskii