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

110After a permutation of the coordinates, we may assume that M ′ is givenin the coordinates t ′ = (z ′ , w ′ ) ∈ C m × C d by the real defining equationsIm w j ′ = ϕ′ j (z′ , ¯z ′ , Re w ′ ), j = 1, . . ., d, where the functions ϕ ′ j are algebraicand vanish at the origin. Solving in terms of w ′ by means of the algebraicimplicit function theorem, we can represent M ′ by the algebraic complexdefining equations(3.22) w ′ j = Θ′ j (z′ , ¯z ′ , ¯w ′ ), j = 1, . . .,d,where Θ ′ satisfies the vectorial functional equation w ′ ≡Θ ′ (z ′ , ¯z ′ , Θ ′ (¯z ′ , z ′ , w ′ )) (which we shall not use). According to thesplitting (z ′ , w ′ ) of coordinates, it is convenient to modify our previous notationby writing z k = f k ′(z′ k ), k = 1, . . .,m and w j = g j ′(w′ j ), j = 1, . . ., dinstead of t i = h ′ i (t′ i ), i = 1, . . .,n, and also{X′k = a ′ k(3.23)(z′ k ) ∂ z k ′ , k = 1, . . .,m, a′ k (0) = 1,Y j ′ = b′ j (w′ j ) ∂ w j ′ , j = 1, . . .,d, b′ j (0) = 1,instead of X i ′ = c′ i (t′ i ) ∂ t ′ . The relation c′ i i (t′ i ) h′ i,t i(t ′ ′ i ) ≡ 1 rewrites down inthe form{ a′k (z k) ′ f ′ k,z k ′ (z′ k) ≡ 1,(3.24)b ′ j (w′ j ) g′ j,w ′ j (w′ j ) ≡ 1.We remind that the derivatives of the f k ′ and of the g′ j are algebraic. Let nowt ′ = (z ′ , w ′ ) ∈ M ′ , thus satisfying (3.22). Then h ′ (t ′ ) = (f ′ (z ′ ), g ′ (w ′ ))belongs to M, namely we have for j = 1, . . ., d:(3.25)g ′ j (w′ j ) − ḡ′ j ( ¯w′ j )2i( f′= ϕ 1 (z 1 ′ ) − ¯f 1 ′(¯z′ 1 )j , . . ., f m ′ (z′ m ) − ¯f m ′ (¯z′ m ) ),2i2iwhere i = √ −1 here. Replacing w ′ j by Θ′ j (z′ , ¯z ′ , ¯w ′ ) in the left hand side,we get the following identity between converging power series of the 2m+dcomplex variables (z ′ , ¯z ′ , ¯w ′ ):(3.26)g ′ j(Θ ′ j(z ′ , ¯z ′ , ¯w ′ )) − ḡ ′ j( ¯w ′ j)2i( f′≡ ϕ 1 (z 1) ′ − ¯f 1(¯z ′ 1)′j , . . ., f m(z ′ m) ′ − ¯f )m(¯z ′ m)′ .2i2iLet us differentiate this identity with respect to z k ′ , for k = 1, . . ., m. Takinginto account the relations (3.24), we obtain(3.27)a ′ k (z′ k ) Θ′ j,z ′ k (z′ , ¯z ′ , ¯w ′ )b ′ j (Θ′ j (z′ , ¯z ′ , ¯w ′ ))≡ ∂ϕ (j f′1 (z 1 ′) − ¯f 1 ′(¯z′ 1 )∂y k 2i, . . ., f m ′ (z′ m ) − ¯f m ′ (¯z′ m ) ).2i