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

218p and has a complex defining equation of the form (7.28) in some systemof local holomorphic coordinates t = (z, w) ∈ C n × C vanishing at p, ispseudospherical at p if and only if its complex graphing function Θ satisfiesthe following explicit nonlinear fourth-order system of partial differentialequations:0 ≡n+1Xn+1Xµ=1 ν=1− δ k 1 ,l 1n+2− δ k 1 ,l 2n+2− δ k 2 ,l 1n+2− δ k 2 ,l 2n+2»j∆ µ [0 1+l1 ] · ∆ν [0 1+l2 ] ∆ ·nX∆ µ [0 1+l ′] · ∆ν [0 1+l2 ]l ′ =1nX∆ µ [0 1+l1 ] · ∆ν [0 1+l ′]l ′ =1nX∆ µ [0 1+l ′] · ∆ν [0 1+l2 ]l ′ =1nX∆ µ [0 1+l1 ] · ∆ν [0 1+l ′]l ′ =11j∆ ·j∆ ·j∆ ·j∆ ·+· ˆδ (n+1)(n+2) k1 ,l 1δ k2 ,l 2+ δ k2 ,l 1δ k1 ,l 2˜·nX nXj· ∆ µ [0 1+l ′] · ∆ν [0 1+l ′′] ∆ ·l ′ =1 l ′′ =1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z l ′∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z l ′∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z l ′′∂t µ∂t µ t ν ] ·ντ=1ff∂ 3 Θ∂z k1 ∂z k2 ∂t τ −ff∂ 3 Θ∂z l ′∂z k2 ∂t τ −ff∂ 3 Θ∂z l ′∂z k2 ∂t τ −ff∂ 3 Θ∂z k1 ∂z l ′∂t τ −ff∂ 3 Θ∂z k1 ∂z l ′∂t τ +ff∂ 3 Θ∂z l ′∂z l ′′∂t τ ,for all pairs of indices (k 1 , k 2 ) with 1 k 1 , k 2 n, and for all pairs ofindices (l 1 , l 2 ) with 1 l 1 , l 2 n.The written system is effective: no implicit formal expression is involvedand pseudosphericality is characterized directly and only in terms of Θ.Now, here is a summarized description of our arguments of proof. Abit similarly as for the C 2 -case — but with major differences afterwards —which was already studied in [21], we may associate to any such Levinondegenerate real analytic local hypersurface M ⊂ C n+1 of equationw = Θ(z, z, w) a uniquely defined system of second-order partial differentialequations:(7.28) w zk1 z k2(z) = Φ k1 ,k 2(z, w(z), wz (w) ) (1 k 1 , k 2 n)with Φ k1 ,k 2= Φ k2 ,k 1, simply by eliminating the two variables z and w,viewed as parameters, from the set of n + 1 equations 23 :w(z) = Θ ( z, z, w ) ( ) ( ), w z1 (z) = ∂Θ∂z 1z, z, w , . . ...., wzn (z) = ∂Θ∂z nz, z, w ,— the assumption that the Jacobian determinant ∆ is nonvanishing atthe origin being precisely the one which guarantees, technically speaking,that the classical (holomorphic) implicit function theorem applies — and23 This process appears for instance in the references [8, Ha1937, Ch1975, Su2001,Su2002, 1, 19].