By Huang X., Yin W.
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Extra resources for A Bishop surface with a vanishing Bishop invariant
For 0 < u 1, define L nor 12 (u) and L 12 (u) N N ) approximates Mnor up to in a similar way as for L 12 . 5 (C), we have L ∗nor 12 (u) = L 12 (u) + O(u −1 N 0
56) for 0 < u 1. 14. 11 as follows. We notice that √ √ (i) σ ∗ (ζ, u) has a convergent power series expansion in (ζ, u) near (0, 0),√ 1 (ii) Ψ(τ, u)√has a convergent power series expansion in τ and u 2s and, √ z −1 √z (iii) σ ( u , u) has a convergent power series expansion in ( √u , u), too. Write Ψ(τ, √ u) = ∞ β aαβ τ α u 2s ∞ and Ψ(τ, Y1 ) = α,β=0 β aαβ τ α Y1 . 57) is analytic in X, Y1 , Y2 near 0. Write ∞ H(X, Y1 , Y2 ) = α,β,γ =0 β γ bαβγ X α Y1 Y2 . Y1 Y2 . 59) Hence, we have |bαβγ | ≤ C0 · Rα+β+γ for a certain positive number C0 and a certain R 1.
23) to conclude that each aα0 β0 with α0 +β0 ≥ N is determined by bαβγ and aαβ with α+β ≤ N −1 through rational functions in aαβ (α + β ≤ N − 1) and bαβγ (α + β + γ ≤ d) with at most C(k0 , d, N ) variables, here C(k0 , d, N ) depends only on k0 , d, N. 22) can be used in the same manner to show that the interior of the closure of J(AdB (n, m; H0 , δ)) ∩ S R in S R is empty. It is easy to see that J(A B ) can be written as a countable union of these sets. We see that J(A B ) is a set of the first category in S R .