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Equations Différentielles et Systèmes de Pfaff dans le Champ by Boele Braaksma (auth.), Raymond Gérard, Jean-Pierre Ramis

By Boele Braaksma (auth.), Raymond Gérard, Jean-Pierre Ramis (eds.)

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Extra info for Equations Différentielles et Systèmes de Pfaff dans le Champ Complexe

Example text

Now ~(m) and S / ~ m such that , hence g is isomorphic to is different From X-S X . Therefore there exists an i s o m o r p h i s m ~(n) M o r e o v e r this i s o m o r p h i s m can be @(n) • @(m) m e r o m o r p h i e in In p a r t i c u l a r S ~(m)/x_ s - ~2/x_ s • chosen in such a w a y that the bases of correspond to the bases of ~ has a global basis m e r o m o r p h i c in V be a (holomorphic) @2 m e r o m o r p h i c in S . 3. Connections. Let V induces dz [1], g . C-linear m o r p h i s m s q d (c£.

D (A,B) = I ; (O,C))-I(XNI(I,Y') we have , ~d hence which must therefore must have chain of natural numbers stop. o. So arrived at the operator where the chain stops, otherwise natural n u m b e r as intersection For completeness we we could again make an operator with a smaller multiplicity, § 3o - INTERSECTION M U L T I P L I C I T Y . a contradiction. OF ANALYTIC BRANCHES. sake we recall in this paragraph the main properties o£ the intersection number. When F and G are (affine) plane curves the intersection number o9 F and G 54 c a n be d e f i n e d (we r e f e r to [2]).

Let T = where mE2Z. C:m :) Then r, = T-IrT+T dT -I ~ = CzY l l + zm z--mY121 TM Y21 Y22 z=O . 38 From the lemma immediately -v(y~j) ~ 1 for all i,j 6 [I,2] So we may assume of the lemma. with XEC follows that that we can choose in such a way that m and then we are ready. -v(Y11 ) = n > 1 and that (Yij) has the form (*) Let and ~E~ . Then Y11 - Xz4Y21 ~ 2 + kz£(Y11 - Y22 ) - k2z24Y21 + k£z'~-I / l F' = Y21 Choose ~ = a11 a12 - a21 a22 Y41 = Yll - kz4Y21 in and ~ = n21 -n11 = n22 - n12 " Look at the pole o£ z=O .