Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

field X 1 ′ := h ∗ (X 1 ). Taking again Theorem 2.1 for granted and proceedingas in the first step of the proof of Proposition 3.1, we may algebraicallystraighten the complex foliation induced by X 1 ′ to the “vertical” foliationby w ′ -lines. Equivalently, we may assume that X 1 ′ = b′ (z ′ , w ′ ) ∂ w ′ with b ′algebraic and b ′ (0) = 1. The assumption h ∗ (∂ w ) = b ′ (z ′ , w ′ ) ∂ w ′ yields thatf ′ (z ′ , w ′ ) is independant of w ′ and that b ′ (z ′ , w ′ ) g w ′ ′(z′ , w ′ ) ≡ 1, so that asin (3.24) above, the derivative gw ′ is algebraic. Let ′w′ = Θ ′ (z ′ , ¯z ′ , ¯w ′ ) bethe complex defining equation of M ′ in these coordinates. The assumptionh ′ (M ′ ) = M yields the following power series identity(3.35) g ′ (z ′ , Θ ′ (z ′ , ¯z ′ , ¯w ′ )) − ḡ ′ (¯z ′ , ¯w ′ ) ≡ 2i ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )).By differentiating this identity with respect to z k ′ , we get(3.36)∂ z ′kg ′ (z ′ , Θ ′ (z ′ , ¯z ′ , ¯w ′ ))+ ∂ z k ′ Θ′ (z ′ , ¯z ′ , ¯w ′ n−1)b ′ (Θ ′ (z ′ , ¯z ′ , ¯w ′ )) ≡ 2i ∑l=1113∂ zl ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) ∂ z ′kf ′l(z ′ ).We notice that the second term in the left hand side of (3.36) is algebraic.By differentiating in turn (3.36) with respect to ¯z k ′ and using the algebraicityof ∂ 2 zg ′ (z ′ , ¯Θ ′ (z ′ , ¯z ′ , ¯w ′ )), we obtain that there exist algebraic functionsk ′ w′A k ′1 ,k 2(z ′ , ¯z ′ ) such that(3.37)A ′k 1 ,k 2(z ′ , ¯z ′ ) ≡∑n−1l 1 ,l 2 =1∂ 2 z l1 ¯z l2ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) ∂ z ′k1f ′l 1(z ′ ) ∂¯z ′k2¯f′l2(¯z ′ ).Without loss of generality, we may assume that h ′ is tangent to the identitymap at t ′ = 0. Then setting ¯z ′ := 0 in (3.37) and using the fact that∂z 2 l1 ¯z l2ϕ(z, 0) = δ l 2l1ε l1 + ∂z 2 l1 ¯z l2χ(z, 0) ≡ δ l 2l1ε l1 by the properties of χ in′Theorem 1.5 we get, since ∂¯z ′k2¯f l2(0) = δ k 2l 2:(3.38) A ′k 1 ,k 2(z ′ , 0) ≡ ε k2 ∂ z ′k1f ′ k 2(z ′ ),which shows that all the first order derivatives ∂ z ′kf l ′(z′) are algebraic.Next, since the canonical transformation to normalizing coordinates is algebraicand preserves the “horizontal” coordinates z ′ (cf. [CM1974]), hencedoes not perturb the complex foliation induced by X 1, ′ we may also assumethat M ′ is given in normal coordinates, namely that the function Θ ′ satisfiesΘ ′ (0, ¯z ′ , ¯w ′ ) ≡ Θ ′ (z ′ , 0, ¯w ′ ) ≡ ¯w ′ . Since the coordinates are normal forboth M and M ′ , it follows by setting ¯z ′ := 0 and ¯w ′ := 0 in (3.35) thatg ′ (z ′ , 0) ≡ 0. Consequently, ∂ z ′kg ′ (z ′ , 0) ≡ 0. Finally, by setting z ′ := 0 and¯w ′ := 0 in (3.36), we see that the first term in the left hand side vanishes andthat the second term is algebraic with respect to ¯z ′ , so we obtain that there