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We construct a complete convergent normal form for a real hypersurface in$\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the firstconvergent normal form for a Levi-degenerate hypersurface. In particular, weobtain, in the spirit of the work of Chern and Moser \cite{chern},distinguished curves in the Levi degeneracy set, that we call \it degeneratechains.

Convergent normal form for real hypersurfaces at generic Levi degeneracy