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1. Hint: Use that there exists an oriented orthonormal base e 1 , . . 10 Choose an orthonormal basis X l , Yl = I (xl ) , . . , xn , Yn = I(xn ) of an euclidian vector space V endowed with a compatible almost complex structure I.
A = 0. iii) Let 0 -:f- a E pk , k :::; n and 0 < i minimal with L i a = 0. 28 one finds 0 = [Li , A] (a) = i(k-n+i- 1)L i - l a and, therefore, k - n + i - 1 = 0. In particular, L n - k (a) -:/- 0. Moreover, L n - k + l a = 0, which will be used in the proof of v) . Assertion iv) follows from i) , ii) , and iii) . v) We have seen already that pk C Ker(Ln - k + l ) . Conversely, let a E 1\ k V * with Ln - k+ l a = 0. Then L n - k + 2 Aa = L n - k + 2 Aa - ALn - k + 2 a = (n - k + 2)Ln - k+l a = 0.
The following local result enables us to define the irreducible decomposition of the zero set Z(f) with multiplicities. 8). Let f E Ocn , o be irreducible. Then for sufficiently small and z E Be: (0) the induced element f E Ocn , z is irreducible. If f, g E Ocn , o are relatively prime, then they are relatively prime in Ocn , z for z in a sufficiently small neighbourhood of 0. Proposition 1 . 35 E 1 Local Theory 22 Proof. We may assume that f E Ocn-t ,0[zi] is a Weierstrass polynomial. Suppose that f as an element of Ocn ,z is reducible.