We show that for any real-analytic submanifold M in ℂN there is a proper real-analytic subvariety V ⊂ M such that for any p ∊ M \ V, any realanalytic submanifold M′ in ℂN, and any p′ ∊ M′, the germs (M, p) and (M′, p′) of the submanifolds M and M′ at p and p′ respectively are formally equivalent if and only if they are biholomorphically equivalent. As an application, for p ∊ M \ V, the problem of biholomorphic equivalence of the germs (M, p) and (M′, p′) is reduced to that of solving a system of polynomial equations. More general results for k-equivalences are also stated and proved.
"Equivalences of Real Submanifolds in Complex Space." J. Differential Geom. 59 (2) 301 - 351, October, 2001. https://doi.org/10.4310/jdg/1090349430