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

Then by differentiating with respect to y x l each one of the following n + 1identically satisfied equations:y ≡ Q ( x 1 ,... ,x n , A 1( x 1 ,... ,x n ,y,y x 1,... ,y x n),...,A n( x 1 ,... ,x n ,y,y x 1,...,y x n),An+1 ( x 1 ,... ,x n )),y,y x 1,...,y x ny x k ≡ Q x k(x 1 ,... ,x n , A 1( x 1 ,... ,x n ,y,y x 1,...,y x n),... ,227A n( x 1 ,... ,x n ,y,y x 1,... ,y x n),An+1 ( x 1 ,... ,x n ,y,y x 1,... ,y x n)),we get the following n + n 2 equations:0 ≡ Q a 1 ∂A1 + · · · + Q ∂y a n ∂An + Q ∂An+1x l ∂y an+1 x l ∂y x lδ k,l = Q x k a 1 ∂A1 + · · · + Q ∂y x x l k a n ∂An + Q ∂y x x l k ∂An+1an+1 ∂y x l(k, l =1··· n).Fixing any l ∈ {1, . . ., n}, thanks to the assumption (Levi nondegeneracy)that the Jacobian determinant:∣ Q□ = □ ( a 1 · · · Q a n Q a n+1 ∣∣∣∣∣∣∣a 1 | · · · |a n |a n+1) Q:=x 1 a 1 · · · Q x 1 a n Q x 1 a n+1. . ..,. .∣Q x n a 1 · · · Q x n a n Q x n a n+1does not vanish, we may solve — just by means of Cramer’s rule — for then + 1 unknowns ∂Aµ , the above system of n + 1 equations, and this gives∂y x lus:(7.28)∂A µ∂y x l= □ [0 µ 1+l ]□ := □(a 1| · · · |a µ−1 |0 1+l |a µ+1 | · · · |a n+1 )□(a 1 | · · · |a µ−1 |a µ |a µ+1 | · · · |a n+1 ) ,where 0 1+l is a specific notation to denote the column consisting of n + 1zeros piled up, except at the (1 + l)-th level from its top, where instead of 0,one reads 1, and where, as our notation with vertical bars helps to guess::=∣□ µ [0 1+l ] = □(a 1| · · · |a µ−1 | µ 0 1+l |a µ+1 | · · · |a n+1 ) :=∣Q a 1 · · · Q a µ−1 0 Q a µ+1 · · · Q a n+1 ∣∣∣∣∣∣∣∣∣∣∣Q x 1 a 1 · · · Q x 1 a µ−1 0 Q x 1 a µ+1 · · · Q x 1 a n+1·· · · · ·· ·· ·· · · · ··Q x k a 1 · · · Q x k a µ−1 1 Q x k a µ+1 · · · Q .x k a n+1·· · · · ·· ·· ·· · · · ··Q x n a 1 · · · Q x n a µ−1 0 Q x n a µ+1 · · · Q x n a n+1To avoid any ambiguity, we shall sometimes put the integer µ in the upperindex position of the vertical bar to indicate precisely which column is concerned.As is clear, this notation allows one to view and to remember what