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

p = (w p , z p , ζ p , ξ p ) ∈ M and for an arbitrary complex “multitime” parameterz 1 = (z 1,1 , . . .,z 1,m ) ∈ C m , we define(5.3) {Lz1 (z p , w p , ζ p , ξ p ) := exp(z 1 L )(p) := exp(z 1,1 L 1 (· · ·(exp(z 1,m L m (p))) · · ·)) :=:= (z p + z 1 , Θ(z p + z 1 , ζ p , ξ p ), ζ p , ξ p ).With this formal definition, there exists a maximal connected open subset Ωof M × C m containing M × {0} such that L z1 (p) ∈ M for all (z 1 , p) ∈ Ω.Analogously, for (ζ 1 , p) running in a similar open subset Ω, we may alsodefine the map(5.4) L ζ1(z p , w p , ζ p , ξ p ) := (z p , w p , ζ p + ζ 1 , Θ(ζ p + ζ 1 , z p , w p )).We notice that the two maps given by (5.3) and (5.4) are holomorphic in theirvariables. Since M is real algebraic, they are moreover complex algebraic.5.2. Segre chains. Let us start from the point p being the origin and let usmove alternately in the direction of S or of S , namely we consider the twomaps Γ 1 (z 1 ) := L z1 (0) and Γ 1 (z 1 ) := L z1(0). Next, we start from theseendpoints and we move in the other direction, namely, we consider the twomaps(5.5) Γ 2 (z 1 , z 2 ) := L z2(L z1 (0)), Γ 2 (z 1 , z 2 ) := L z2 (L z1(0)),where z 1 , z 2 ∈ C m . Also, we define Γ 3 (z 1 , z 2 , z 3 ) := L z3 (L z2(L z1 (0))),etc. By induction, for every positive integer k, we obtain two mapsΓ k (z 1 , . . .,z k ) and Γ k (z 1 , . . ., z k ). In the sequel, we shall often use the notationz (k) := (z 1 , . . .,z k ) ∈ C mk . Since Γ k (0) = Γ k (0) = 0, for everyk ∈ N ∗ , there exists a sufficiently small open polydisc ∆ mk (δ k ) centered atthe origin in C mk with δ k > 0 such that Γ k (z (k) ) and Γ k (z (k) ) belong to Mfor all z (k) ∈ ∆ mk (δ k ).We also exhibit a simple link between the maps Γ k and Γ k . Let σ bethe antiholomorphic involution defined by σ(t, τ) := (¯τ, ¯t). Since w =Θ(z, ζ, ξ) if and only if ξ = Θ(ζ, z, w), this involution maps M onto Mand it also fixes the antidiagonal Λ pointwise. Using the definitions (5.3)and (5.4), we see readily that σ(L z1 (0)) = L ¯z1 (0). It follows generally thatσ(Γ k (z (k) )) = Γ k (z (k) ).Next, we observe that Γ k+1 (z (k) , 0) = Γ k (z (k) ), since L 0 and L 0 coincidewith the identity map. So the ranks at the origin of the maps Γ k increase withk.Definition 5.1. The real analytic generic manifold M is said to be minimalat p if the maps Γ k are of (maximal possible) rank equal to 2m + d =dim C M at the origin in ∆ mk (δ k ) for all k large enough.The following fundamental properties are established in [Me1998],[Me2001].121