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

236From (1.12), the map ¯w ↦→ Θ(0, 0, ¯w) is already of rank m at ¯w = 0.One then verifies ([BER1999, Me2005a, Me2005b, MP2005]) that there existmultiindices β(1), . . ., β(n) ∈ N n with |β(k)| 1 for k = 1, . . .,nand max 1kn |β(k)| = κ together with integers j(1), . . ., j(n) with 1 j(k) m such that the local holomorphic map(1.15) ( (ΘC n+m j ) ( ) )1jm,∋ (¯z, ¯w) ↦−→ (0, ¯z, ¯w) Θ j(k)z β(k)(0, ¯z, ¯w) ∈ C m+n1knis of rank n + m at (¯z, ¯w) = (0, 0).1.16. Associated system of partial differential equations. Generalizingan idea which goes back to B. Segre in [Se1931, Se1932] (n = m = 1),applied by É. Cartan in [Ca1932a] and studied more recently in [Su2001,GM2003a], we may associate to M a system of partial differential equationsof the form (E ) as follows. Complexifying the variables ¯z and ¯w, weintroduce new independent variables ζ ∈ C n and ξ ∈ C m together with thecomplex algebraic or analytic m-codimensional submanifold M of C 2(n+m)defined by(1.17) w j = Θ j (z, ζ, ξ), j = 1, . . ., m.We then consider the “dependent variables” w j as algebraic or analytic functionsof the “independent variables” z k , with additional dependence on theextra “parameters” (ζ, ξ). Then by applying the differentiation ∂ |α| /∂z αto (1.17), we get w j z α(z) = Θj zα(z, ζ, ξ). Assuming finite nondegeneracyand writing these equations for (j, α) = (j(k), β(k)), we obtain a system ofm + n equations:⎧⎨ w j (z) = Θ j (z, ζ, ξ), j = 1, . . ., m,(1.18)⎩w j(k)(z) = Θ j(k)z β(k) zβ(k)(z, ζ, ξ),k = 1, . . .,n.By means of the implicit function theorem we can solve:(1.19) (ζ, ξ) = R ( z k , w j (z), w j(k)z β(k) (z) ) .Finally, for every pair (j, α) different from (j, 0) and from (j(k), β(k)),we may replace (ζ, ξ) by R in the differentiated expression w j z α(z) =Θ j zα(z, ζ, ξ), which yields(w j z α(z) = Θj z z, R ( z k , w j (z), w j(k) (z) ))α z(1.20)(β(k) )=: Fαj z k , w j (z), w j(k) (z) .z β(k)This is the system of partial differential equations associated to M.