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35) X. Huang, W. Yin Now, let F : M → M be a formal equivalence map with F := ( f , g ) = (z, w) + (O(|w| + |z|2 ), O(w2 )) and let the polynomial map F( N+1) be the Taylor polynomial of F of order N, as before. Here N = Ns + s − 1. Then F( N+1) (M) approximates M up to order N. 8, we get L ∗12 (g( N+1) (u)) = L 12 (u) + O(u N−2 ). 36) Here, as before, the polynomial g( N+1) (u) is the Taylor polynomial of g(u) at the origin of order N. We mention again that if φ is a formal power series 1 in u 2s and h(u) is a formal power series in u without constant term, then 1 φ ◦ h gives a formal power series in u 2s .

8. We first assume that M, M are already normalized up to order N. 2, we see that F = Id + O(|(z, w)| N ), M is defined by w = zz + 2Re{ϕ0 (z)} + o(|z| N ), M is defined by w = zz + 2Re{ϕ0 (z)} + o(|z| N ), where ϕ0 (z) = z s + o(z s ), u = u + g(u) = (sk+ j ) u + O(|u| N ) and ϕ0 (0) = 0 for j = 0, 1 mod s. Since u = u + g(u) = u + O(u N ) and u = r 2 , u = r 2 , we have r = r + O(u N−1 ). From the way σ and σ ∗ were constructed, we claim that there is a constant C independent of τ and u such that for 0 < u 1, we have the following: |σ ∗ (τ, r ) − σ(τ, r)| ≤ C|τ|u N−1 for τ ∈ ∆.

Here, f ( N+1) is the Taylor polynomial of order N th in the Taylor expansion of f at 0, as defined before. Write M and M for the local holomorphic hull of M and M near the origin, respectively. We next construct a holomorphic map from M \ M to M \ M as follows: Let Ψ(·, r) be the biholomorphic map from ∆ to itself such that Ψ(τ j (u), r) = τ ∗j (u) for j = 0, 1. Since τ j (u), τ ∗j (u) ∈ ∆, to see the existence and uniqueness of Ψ(·, r), it suffices for us to explain that dhyp (τ0 (u), τ1 (u)) = dhyp (τ0∗ (u), τ1∗ (u)).

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