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For any real-analytic hypersurface M in complex euclidean space of dimension>= 2 which does not contain any complex-analytic subvariety of positivedimension, we show that for every point p in M the local real-analytic CRautomorphisms of M fixing p can be parametrized real-analytically by theirl(p)-jets at p. As a direct application, we derive a Lie group structure forthe topological group Aut(M,p). Furthermore, we also show that the order l(p)of the jet space in which the group Aut(M,p) embeds can be chosen to dependupper-semicontinuously on p. As a first consequence, it follows that that givenany compact real-analytic hypersurface M in complex euclidean space, thereexists an integer k depending only on M such that for every point p in M germsat p of CR diffeomorphisms mapping M into another real-analytic hypersurface ina complex space of the same dimension are uniquely determined by their k-jet atthat point. Another consequence is a boundary version of H. Cartan's uniquenesstheorem.  Our parametrization theorem also holds for the stability group of anyessentially finite minimal real-analytic CR manifold of arbitrary codimension.One of the new main tools developed in the paper, which may be of independentinterest, is a parametrization theorem for invertible solutions of a certainkind of singular analytic equations, which roughly speaking consists ofinverting certain families of parametrized maps with singularities.Comment: to appear in J. Amer. Math. So

Parametrization of local CR automorphisms by finite jets and applications