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

in K[a, z] n+m and in K[x, c] p+m . We notice that, whereas ϕ and h are apriori ( only purely formal, by construction, Φ 0 0 and H0 0 are K-analytic near0, 0, Jκ ∗c h(0)) and near ( 0, 0, Jz κϕ(0)) .Next, we introduce the following vector fields with K-analytic coefficientstangent to M :⎧V ⎪⎨ j := ∂∂y + ∑ m∂Π ∗l ∂(a, z) j ∂yj ∂bl, j = 1, . . .,m and(11.22)⎪⎩l=1V ∗ j := ∂∂b j + m∑l=1∂Π l ∂(x, c)∂bj ∂y l,Indeed, we check that V j1 [b j 2− Π ∗j 2(a, z)] ≡Π j 2(x, c)] ≡ 0.j = 1, . . .,m.3010 and that V ∗ j 1[y j 2−For δ ′ ∈ N m , we observe that V δ′ ϕ = ∂|δ′| ϕ. Applying then L β′ with∂y δ′β ′ ∈ N n , we get for i = 1, . . .,n + m:((11.23) L β′ V δ′ ϕ i (z) = Q β ′ ,δ ′ z, c, J|β ′ |+|δ ′ |z ϕ i (z) ) ,with Q β ′ ,δ ′ universal. Since the n + m vector fields L k and V j , having coefficientsdepending on (z, c), span the tangent space to K n x × Km y , the changeof basis of derivations yields, by induction, the following.Lemma 11.24. For every α ∈ N n+m , there exists a universal polynomial P αin its last variables with coefficients being K-analytic in (z, c) and dependingonly on Π, Π ∗ such that, for i = 1, . . ., n + m:(11.25) ∂z α ϕi (z) ≡ P α(z, c, ( L β′ V δ′ ϕ i (z) ) ).|β ′ |+|δ ′ ||α|We are now in position to state and to prove the first fundamental technicallemma which generalizes the two formulas (11.20) to arbitrary jets.Lemma 11.26. For every λ ∈ N, there exist two local K-analytic maps, Φ λ 0valued in K (n+m)Cλ n+m+λ , and Hλ0 valued in K (p+m)Cλ p+m+λ , such that:{ (Jλz ϕ(z) ≡ Φ λ 0 z, c, Jκ ∗ +λc h(c) ) ,(11.27)Jc λ h(c) ≡ ( Hλ 0 z, c, Jκ+λz ϕ(z) ) .Proof. Consider for instance the first line. To obtain it, it suffices to applythe derivations L β′ V δ′ with |β ′ | + |δ ′ | λ to the first line of (11.20), to usethe chain rule and to apply Lemma 11.24.Let θ ∈ K l , l ∈ N, let Q(θ) = ( Q 1 (θ), . . .,Q n+2m+p (θ) ) ∈ K[θ] n+2m+pand let a 1 ∈ K p . As the multiple flow of L ∗ given by (10.10) does not act onthe variables (x, y), we have the trivial but crucial property:(11.28)ϕ ( L ∗ a 1(Q(θ)) ) ≡ ϕ ( π z (L ∗ a 1(Q(θ))) ) ≡ ϕ (π z (Q(θ))) ≡ ϕ (Q(θ)) .