By Huang X., Yin W.
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Extra info for A Bishop surface with a vanishing Bishop invariant
We next give the precise defining equation of B near 0. 5) where h 1 (ζ) is holomorphic near 0. 3), we get w = h 2 (ζ) = (1 − s)ζ s + o(ζ s ). 6), we get − w = (h 3 (ζ))s s−1 with h 3 (ζ) = ζ + o(ζ). 7) Hence, we get ζ= w − s−1 h −1 3 z = h1 ◦ h −1 3 = s(−1)− s w 1 w − s−1 s−1 s 1 s = (−1) 1 s · (s − 1) 1−s s 1 s 1 s−1 = h 1 (−1) + o(w s−1 s ). 8) X. Huang, W. Yin Here, h j s are holomorphic functions near 0. 8). Next, let w = u ≥ 0 and we define e− Aj (u) = h 1 ◦ h −1 3 = se √ (1+2 j )π −1 s u √ (2 j+1)π −1 s s−1 s u s−1 · (s − 1) 1−s s 1/s + o(u s−1 s ), j = 0, 1, .
17) Here we assume that F = ( f (z, w), g(z, w)) = (z + f(z, w), w+g(w)) with f(z, w) = O(|w| + |z|2 ), g(w) = O(w2 ) and g(w) = g(w). (2) Suppose that there is a formal holomorphic map F : M → M , where we write F = ( f (z, w), g(z, w)) = (z + f(z, w), w + g(w)) with f(z, w) = O(|w| + |z|2 ) and g(w) = O(w2 ). For an N > 1, write, for the rest of this paper, f ( N+1) (z, w), g( N+1) (z, w) for the (Taylor) polynomials consisting of terms of degree ≤ N in the Taylor expansions at the origin of f and g, respectively, with N = Ns + s − 1.
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 . 36) the following: L ∗12 (g(u)) = L 12 (u) in the formal sense. 37) have the same formal power series expansion in u 1/(2s) .
A Bishop surface with a vanishing Bishop invariant by Huang X., Yin W.