Let M ⊂ Cn+1 (n ≥ 2) be a real analytic submanifold defined by an equation of the form: w =|z | 2 + O(|z | 3), where we use (z, w) ∈ Cn × C for the coordinates of Cn+1. We first derive a pseudonormal form for M near 0. We then use it to prove that (M, 0) is holomorphically equivalent to the quadric (M ∞ : w =|z | 2, 0) if and only if it can be formally transformed to (M∞, 0). We also use it to give a necessary and sufficient condition when (M, 0) can be formally flattened. Our main theorem generalizes a classical result of Moser for the case of n = 1. Downloaded fro
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