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

298Observe that differentiating the first line of (11.5) with respect to x k amountsto applying the derivation L k . Similarly, differentiating the third lineof (11.5) with respect to a q amounts to applying L ∗ q . We thus get for(z, c) ∈ M(11.9)⎧L ⎪⎨ k ψ(z) =⎪⎩n∑l=1L ∗ q g(c) = p∑r=1∂Π ′ ( )∂x ′ l φ(z), h(c) Lk φ l (z) and∂Π ′ ∗∂a ′ r(f(c), ϕ(z))L∗q f r (c),It follows from det ( ) (∂ϕ k∂z (0) ≠ 0 and det ∂h k) l ∂c (0) ≠ 0 that the two formalldeterminants(11.10) det ( L k φ l (z) ) 1ln1knand det ( L ∗ q fr (c) ) 1rp1qphave nonvanishing constant term. Consequently, these two matrices are invertiblein K[z] and in K[c]. So there exist universal polynomials S j landS ∗j r such that⎧(∂Π ′ j( ) S j {Lk′lϕ i′ (z) } )1i ′ n+m1k⎪⎨ ∂x ′ l ϕ(z), h(c) = ′ ndet ( L k ′ φ l′ (z) ) and1l ′ n1k(11.11)′ n( {L∂Π ′ ∗j( ) S ∗j ∗r q ′ h i′ (c) } )1i ′ p+m1q⎪⎩ ∂a ′ r f(c), ϕ(z) = ′ pdet ( L ∗ qf ′ r′ (c) ) ,1r ′ p1q ′ pfor 1 j m, for 1 l n, for 1 r p and for (z, c) ∈ M .Again, we apply the vector fields L k to the obtained first line and thevector fields L ∗ q to the obtained second line, getting, thanks to the chain rule:(11.12) ⎧( {Lkn∑ ∂ 2 Π ′ j( ) R j ′⎪⎨∂x ′ l 1x ′ l 2φ(z), h(c) Lk φ l l 1 ,k 1L k ′2ϕ i′ (z) } )1i ′ n+m21k 1(z) =′ ,k′ 2[nl 2 =1det ( L k ′ φ l′ (z) ) ]1l ′ n 2and1k ′ n⎪⎩p∑r 2 =1∂ 2 Π ′ ∗j∂a ′ r 1a ′ r 2(f(c), ϕ(z))L∗q f r 2(c) =( {LR ∗j ∗r 1 ,q q 1L ∗ ′ qh2′ i′ (c) } 1i ′ p+m[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2,1q ′ p1q ′ 1 ,q′ 2 p )for 1 j m, for 1 l 1 , l 2 n, for 1 r 1 , r 2 p and for (z, c) ∈ M .Here, R j l 1 ,k and R∗j r 1 ,q are universal polynomials. Then applying once more