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By Huang X., Yin W.

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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 .

1 (iii). 37)) that L ∗12 (g(u)) = L 12 (u) in the formal sense. Write u = V 2s . Define U = (g(u))1/(2s) = u 1/(2s) + . . , which has 1 a formal power series expansion in u 2s and thus can be regarded as a formal power series in V . Then L ∗12 (U 2s (V )) = L 12 (V 2s ) in the formal sense. Notice that L ∗12 (t ∗2s ) and L 12 (t 2s ) have convergent power series expansions in t ∗ and t, respectively. Moreover, L ∗12 (t ∗2s ) = (ψ ∗ (t ∗ ))s−2 , L 12 (t 2s ) = (ψ(t))s−2 A Bishop surface with a vanishing Bishop invariant with ψ, ψ ∗ invertible holomorphic map of (C, 0) to itself, and with ψ (0) = 1 ψ ∗ (0)(= |2(Cs−2,0 − Cs−2,1 )| s−2 ).

9). Here F = ( f , g ) = (z + f, w + g) is assumed to be a holomorphic map with f = O(|w| + |z|2 ), g(z, w) = g(w) = O(w2 ), g(w) = g(w) and u = u + g(u). 3. 2, since it is not assumed that F(M) ⊂ M , the reality of g is not automatic from the property that F(M) approximates M to a high order. 2. 11) k=1 j=2 N N with Mnor defined by and let Φ2 be a biholomorphic map from M to Mnor N s−1 s w = z z + 2Re z + a ks+ j z ks+ j + R (z , z ). 12) k=1 j=2 Here R(z, z) = R(z, z) = o(|z|sN+s−1 ) and R (z, z) = R (z, z) = o(|z|sN+s−1 ).

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