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

Cramer’s rule, we get(11.13) ⎧∂ 2 Π ′ j⎪⎨(∂ 2 Π ′ ∗j( ) S ∗j r 1 ,r 2⎪⎩ ∂a ′ r 1a ′ r 2 f(c), ϕ(z) = [1k ′ 1 ,k′ 2 n )(( ) S j {Lk′l 1 ,l 2 1L k ′2ϕ i′ (z) } 1i ′ n+m∂x ′ l 1x ′ l 2φ(z), h(c) = [det ( L k ′ φ l′ (z) ) ]1l ′ n 3and1k ′ n{L ∗ qL ∗ 1′ q 2′h i′ (c)} 1i′ p+mdet ( L ∗ q ′ f r′ (c) ) 1r ′ p1q ′ p1q ′ 1 ,q′ 2 p )] 3.By induction, for every j with 1 j m and every two multiindicesβ ∈ N n and δ ∈ N p , there exists two universal polynomials S j β and S∗ jδ suchthat⎧( {L∂ |β| Π ′ j( ) S j β ′βϕ i′ (z) } )1i ′ n+m|β⎪⎨∂x ′ β φ(z), h(c) = ′ ||β|[det ( L k ′ φ l′ (z) ) ]1l ′ n 2|β|+1and1k(11.14)′ n(∂ |γ| Π ′ ∗j( ) S ∗ j {L∗δ ′δh i′ (c) } )1i ′ p+m|δ⎪⎩ ∂a ′ δ f(c), ϕ(z) = ′ ||δ|[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2|δ|+1.1q ′ pHere, for β ′ ∈ N n , we denote by L β′ the derivation of order |β ′ | defined by(L 1 ) β′ 1 · · ·(Ln ) β′ n . Similarly, for δ ′ ∈ N p , L ∗δ′ denotes the derivation of order|δ ′ | defined by (L ∗ 1) δ′ 1 · · ·(L∗p ) δ′ p .Next, by the assumption that M ′ is solvable with respect to the parameters,there exist integers j(1), . . .,j(p) with 1 j(q) m and multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 and max 1qp |β(q)| = κ such thatthe local K-analytic map(11.15) ⎛K p+m ∋ c ′ ↦−→ ⎝ ( Π ′j (0, c ′ ) ) () ⎞1jm ∂ |β(q)| Π ′ j(q), (0, c ′ ) ⎠ ∈ K p+m∂x ′ β(q)1qpis of rank p + m at c ′ = 0. Similarly, by the assumption that M ′ is solvablewith respect to the variables, there exist integers j ∼ (1), . . .,j ∼ (n) with1 j ∼ (l) m and multiindices δ(1), . . ., δ(p) ∈ N n with |δ(q)| 1 andmax 1qp |δ(q)| = κ ∗ such that the local K-analytic map(11.16) ((ΠK n+m ∋ z ′ ↦−→′∗j (0, z ′ ) ) ()1jm ∂ |δ(l)| Π ′ ∗j ∼ (l), ∈ K n+m∂a ′ δ(l)(0, z′ )) 1lnis of rank n + m at z ′ = 0. We then consider from the first line of (11.14)only the (p + m) equations written for (j, 0), (j(q), β(q)) and we solve h(c)299