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

108We shall call a CR generic manifold M ′′ having infinitesimal CR automorphismsof the form X i ′′ = c ′′i (t′′ i ) ∂ t ′′ with c ′′i i (0) ≠ 0 a pseudotube. Sucha pseudotube is not in general a product by R n . In fact, there is no hopeto tubify all algebraic peudotubes in algebraic coordinates, as shows the elementaryexample Im w = |z + 1| 2 + |z + 1| 6 − 2 having infinitesimalCR automorphisms ∂ w and i(z + 1)∂ z , since the only change of coordinatesfor which Φ ∗ (∂ w ) = ∂ w ′ and Φ ∗ (i(z + 1)∂ z ) = ∂ z ′ is z + 1 = e iz′ ,w = w ′ , which transforms M into M ′ of nonalgebraic defining equationIm w ′ = e −2y′ + e −6y′ .The constructions of this paragraph may be represented by the followingsymbolic picture.C n t v x, uMyΦC n v t ′′M ′x ′ , u ′y ′M strong tubeM ′ algebraicΨ ′C n t ′′ v ′′ x ′′ , u ′′′Φ ′′ := Ψ ′ ◦ ΦM ′′y ′′M ′′ algebraic pseudotubeSummary and conclusion of the first step. To conclude, let us denote forsimplicity M ′′ again by M ′ , the coordinates t ′′ again by t ′ and t ′′ = Ψ ′ ◦Φ(t) by t ′ = Φ(t). After the above straightenings, we have shown thatthe infinitesimal CR automorphisms X i ′ := Φ ∗(X i ) of the algebraic genericmanifold M ′ are of the sympathetic form X i ′ = c′ i (t′ i ) ∂ t ′ , i = 1, . . ., n, withialgebraic coefficients c ′ i(t ′ i) satisfying c ′ i(0) = 1.3.2. Proof of the second step. We characterize first finite nondegeneracyfor tubes of codimension d in C n .Lemma 3.2. Let M be a tube of codimension d in C n equipped with coordinates(z, w) = (x + iy, u + iv) ∈ C m × C d given by the equationsv j = ϕ j (y), j = 1, . . ., d, where ϕ j (0) = 0. Then M is finitely nondegenerateat the origin if and only if there exist m multi-indices β∗ 1, . . .,βm ∗ ∈ Nm