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

Clearly, the left hand side is an algebraic function A j,k ′ (z′ , ¯z ′ , ¯w ′ ). Then differentiatingagain with respect to the variables z k ′ the relations (3.27), we seethat for every multi-index β ∈ N m with |β| ≥ 1, and every j = 1, . . ., d,there exists an algebraic function A j,β ′ (z′ , ¯z ′ , ¯w ′ ) such that the followingidentity holds:(3.28)A j,β ′ (z′ , ¯z ′ , ¯w ′ ) ≡ ∂β 1+···+β mϕ j∂y β 11 · · ·∂βm( f′1 (z 1) ′ − ¯f 1(¯z ′ 1)′y m2i111, . . ., f m(z ′ m) ′ − ¯f )m(¯z ′ m)′ .2iDifferentiating (3.28) with respect to ¯w ′ , we see immediately that A j,β ′ isin fact independent of ¯w ′ . Furthermore, we see that A j,β ′ is real, namelyA j,β ′ (z′ , ¯z ′ ) ≡ A ′ j,β(¯z ′ , z ′ ). Now we extract from (3.28) the m identitieswritten for β := β∗ k, j := jk ∗ , k = 1, . . .,m and we use the invertibility ofthe mapping ψ defined in (3.19) (recall that ψ ′ (y ′ ) = y denotes the inverseof y ′ = ψ(y)), which yields(3.29)f ′ k (z′ k ) − ¯f ′ k (¯z′ k )2i≡ ψ ′ k (A ′j 1 ∗,β 1 ∗ (z′ , ¯z ′ ), . . .,A ′j m ∗ ,β m ∗ (z′ , ¯z ′ )),for k = 1, . . .,m. For simplicity, we shall write A k ′(z′, ¯z ′ ) instead ofA ′j∗ ∗(z ′ , ¯z ′ ). Finally, we differentiate (3.29) with respect to z ′ k,βk k , whichyields, taking into account (3.24):(3.30) ⎧⎪⎨⎪⎩12i a ′ k (z′ k ) ≡0 ≡m∑l=1m∑l=1∂ψ k′ (A ′∂yl′1 (z′, ¯z ′ ), . . ., A m ′ (z′, ¯z ′ )) ∂A l′(z ′ , ¯z ′ ),∂z ′ k∂ψ k′ (A∂y1(z ′ ′ , ¯z ′ ), . . ., A m(z ′ ′ , ¯z ′ )) ∂A l′(z ′ , ¯z ′ ), ˜k ≠ k.l′ ∂ze ′ kIt follows from these relations (3.30) viewed in matrix form that the constantmatrix ( ∂A l′(0, 0))∂z k′ 1≤l,k≤m is invertible, because the diagonal matrix(δ e kk [2ia ′ k (z′ k )]−1 ) 1≤k, e k≤mis evidently invertible at z k ′ = 0 (recall a′ k (0) = 1).Consequently, there exist algebraic functions B k,l ′ (z′ , ¯z ′ ) so that(3.31)∂ψ k′ (A ′∂yl′ 1 (z′ , ¯z ′ ), . . ., A m ′ (z′ , ¯z ′ )) ≡ B k,l ′ (z′ , ¯z ′ ).Next, setting ỹ k ′ = A k ′(iy′, −iy ′ ), k = 1, . . ., m we see, from the invertibilityof the matrix ( ∂A k′ (0, 0))∂z l′ 1≤k,l≤m and from the reality of A k ′(z′, ¯z ′ ), that theJacobian determinant at the origin of the mapping y ′ ↦→ A ′ (iy ′ , −iy ′ ) = ỹ k′is nonzero. Thus there are real algebraic functions C k ′ so that we can express