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

128l 0 + l, we get(6.18) ⎧J l h(Γ k+1 (z (k+1) )) = Π l (Γ k+1 (z (k+1) ), J l 0+l¯h(Γk+1 (z (k+1) ))) =⎪⎨= Π l (Γ k+1 (z (k+1) ), J l 0+l¯h(Γk (z (k) ))) =⎪⎩= Π l (Γ k+1 (z (k+1) ), Π l0 +l,k(Γ k (z (k) ), J kl 0+l 0 +l¯h(0))) =:=: Π l,k+1 (Γ k+1 (z (k+1) ), J (k+1)l 0+l¯h(0)),which yields the desired formula at level k+1. For the above formal compositionformulas to be correct, we possibly have to shrink ε. Finally, a directinspection of relative polynomialness shows that Π l,k+1 is polynomial withrespect to the jet variables J α i with |α| ≥ (k + 1)l 0 + 1, i = 1, . . .,n. Theproof of Lemma 6.21 is complete.6.4. Algebraic parameterization of CR mappings by their jet at the origin.Finally, as in the paragraph after Theorem 5.2, we choose ρ 4 > 0 sufficientlysmall such that Γ µ0 maps the polydisc ∆ mµ0 (η) submersively ontoan open neighborhood of the origin in M which contains the open subsetM ∩ (∆ n (ρ 4 ) × ∆ n (ρ 4 )). From the relation Γ µ0 +1(z (µ0 ), 0) ≡ Γ µ0 (z (µ0 )),it follows trivially that Γ µ0 +1 also induces a submersion from ∆ m(µ0 +1)(η)onto M ∩(∆ n (ρ 4 )×∆ n (ρ 4 )). It follows that the composition π t ◦Γ µ0 +1 alsomaps submersively the polydisc ∆ m(µ0 +1)(η) onto an open neighborhood ofthe origin in C n which contains ∆ n (ρ 4 ). Consequently, in the representationobtained in Lemma 6.21 with l = 0 and k := µ 0 + 1 = 2ν 0 + 2 (which iseven), namely in the representation(6.19) ¯h(Γµ0 +1(z (µ0 +1))) = Π 0,µ0 +1(Γ µ0 +1(z (µ0 +1)), J (µ 0+1)l 0¯h(0)),we can write an arbitrary t ∈ ∆ n (ρ 4 ) in the form Γ µ0 +1(z (µ0 +1)), and finally,conjugating (6.19), we obtain a complex algebraic mapping H withthe property that h(t) = H(t, J (µ 0+1)l 0h(0)). We may now summarize whatwe have proved so far.Theorem 6.4. Let M be a real algebraic generic submanifold in C n passingthrough the origin, of codimension d ≥ 1 and of CR dimension m = n−d ≥1. Assume that M is l 0 -nondegenerate at 0. Assume that M is minimal at0, let ν 0 be the Segre type of M at 0 and let µ 0 := 2ν 0 + 1 be the Segretype of M at 0. Let κ 0 := (µ 0 + 1)l 0 . Let t = (z, w) ∈ C m × C d beholomorphic coordinates vanishing at 0 with T 0 M = {Im w = 0} andlet ρ 1 > 0 be such that M is represented by the complex analytic definingequations ξ j = Θ j (ζ, t), j = 1, . . ., d in ∆ n (ρ 1 ). Then there exist ε > 0,ρ 4 > 0 and there exists a complex algebraic C n -valued mapping H(t, J κ 0)defined for |t| < ρ 4 and for |J κ 0− J κ 0Id | < ε which satisfies H(t, Jκ 0Id ) ≡ tand which depends only on the defining functions ¯w j − Θ j (¯z, t) of M , such