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

222has nonvanishing (n+1) ×(n+1) Jacobian determinant at (z, w) = (0, 0).It follows then that this Jacobian determinant, not restricted to the origin:(7.28) ∆ = ∆ ( z, z, w ) Θ z1 · · · Θ zn Θ w:=Θ z1 z 1· · · Θ z1 z nΘ z1 w·· · · · ·· ··∣ Θ znz 1· · · Θ znz nΘ znw∣does not vanish in some small neighborhood of the origin in C n × C n × C.Levi nondegeneracy at the central point, i.e. ∆ ≠ 0 locally, will be assumedthroughout the present paper.Associated system of partial differential equations. At least since thepublication in 1888 by Lie and Engel in Leipzig of the Theorie der Transformationsguppen,it is known in a very general context — see Chapter 10of [8] and also [23, Ha1937, Ch1975, Fa1980, 19, 1, 21] — that, to thewhole family of Segre varieties:S z,w := { (z, w) ∈ C n × C: w = Θ ( z, z, w )}parametrized by the n + 1 antiholomorphic variables ( z 1 , . . ., z n , w ) , onemay canonically associate a completely integrable second-order system ofpartial differential equations whose general solution is precisely the functionΘ ( z, z, w ) . Indeed, considering w as a function w = w(z) of (z 1 , . . .,z n )in the defining equation of M, one differentiates it once with respect to eachvariable z 1 , . . .,z n so that one gets the n + 1 equations:w(z) = Θ ( z, z, w ) ,w z1 (z) = ∂Θ∂z 1(z, z, w), . . ...., wzn (z) = ∂Θ∂z n(z, z, w).Then by means of the implicit function theorem — which applies preciselythanks to the nonvanishing of ∆ —, one may clearly solve for the n + 1antiholomorphic “parameters” (z, w), and this procedure provides a representation:z 1 = ζ 1(z, w(z), wz (z) ) , . . ., z n = ζ n(z, w(z), wz (z) ) , w = ξ ( z, w(z), w z (z) )with certain n + 1 uniquely defined local complex analytic functionsζ i (z, w, w z ) and ξ(z, w, w z ) of 2n + 1 complex variables. Utilizing thesefunctions, one is then pushed to replace z and w in all possible second-orderderivative:( )w zk1 z k2(z) =∂2 Θ∂z k1 ∂z k2z, z, w( ((7.28) = ∂2 Θ∂z k1 ∂z k2z, ζ z, w(z), wz (z) ) , ξ ( z, w(z), w z (z) ))=: Φ k1 ,k 2(z, w(z), wz (z) ) (k 1 , k 2 =1··· n),