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

124now differentiate (6.2) by applying the vector fields L 1 , . . .,L m , whichgivesm∑ ∂Θ j(6.3) L k ḡ j (τ) = (∂ζ ¯f(τ), h(t)) L k ¯fl (τ),ll=1for k = 1, . . ., m and j = 1, . . .,d. For fixed j, we consider the m equations(6.3) as an affine system satisfied by the partial derivatives ∂Θ j /∂ζ l .By Cramer’s rule, there exists universal polynomials Ω j,k in their variablessuch that(6.4)∂Θ j∂ζ k( ¯f(τ), h(t)) = Ω j,k({L l ¯h(τ)}1≤l≤m )det (L k ¯fl (τ)) 1≤k,l≤nfor all (t, τ) ∈ M with |t|, |τ| < ρ 2 and for k = 1, . . ., m, j = 1, . . .,d.Applying the derivations L k to (6.4) we see by induction that for everymulti-index β ∈ N m ∗ and for every j = 1, . . .,d, there exists a universalpolynomial Ω j,β in its variables such that(6.5)1 ∂ |β| Θ jβ! ∂ζ ( ¯f(τ), h(t)) = Ω j,β({L γ ¯h(τ)}|γ|≤|β| )β [det (L k ¯fl (τ)) 1≤k,l≤n ] 2|β|−1,for all (t, τ) ∈ M with |t|, |τ| < ρ 2 . Here, for γ = (γ 1 , . . .,γ m ) ∈ N m ,we denote by L γ the derivation (L 1 ) γ 1. . .(L m ) γm . Next, denoting byω j,β (t, τ) the right hand side of (6.5) and developing the left hand side inpower series using (5.7), we may write(6.6) Θ j,β (h(t)) + ∑( ¯f(τ)) γ Θ j,β+γ (h(t)) = ω j,β (t, τ).(6.7)h(t) = Ψ⎪⎨⎪⎩γ∈N m ∗Recall that M is l 0 -nondegenerate at 0. Using (5.8), we can solve h(t) interms of the derivatives of ¯h(τ), namely⎧ (¯f(τ),Ω j 1 ∗ ,β 1 ∗ ({L γ ¯h(τ)}|γ|≤|β 1 ∗ |)[det (L k ¯fl (τ)) 1≤k,l≤n ] 2|β1 ∗ |−1, . . .. . .,Ω j n ∗ ,β n ∗ ({L γ ¯h(τ)}|γ|≤|β n ∗ |)[det (L k ¯fl (τ)) 1≤k,l≤n ] 2|βn ∗ |−1= Ψ( ¯f(τ), ω j 1 ∗ ,β∗ 1(t, τ), . . .,ω j∗ n (t, τ)). ,βn ∗)=Here, the maximal length of the multi-indices β 1 ∗ , . . .,βn ∗ is equal to l 0. Accordingto (5.8), the representation (6.7) of h(t) holds provided |ḡ(τ)| < ˜ρ 3and |ω j i ∗ ,β i ∗ | < ˜ρ 3. Since the coordinates are normal, we have Θ j (¯z, 0, 0) ≡0, or equivalently Θ j,β (0) = 0 for all j = 1, . . .,d and all β ∈ N m . It followsfrom (6.6) and from h(0) = 0 that ω j,β (0) = 0, for all j = 1, . . .,d and allβ ∈ N m . Consequently, there exists a radius ρ 3 ∼ ˜ρ 3 with 0 < ρ 3 < ρ 2 < ρ 1