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

96ψ(y) = y ′ defined by (1.1) is of rank m at the origin in R m y and let y = ψ ′ (y ′ )denote the local inverse in ψ(y). Assume that M ∈ Tn d , namely M is astrong tube of codimension d. If M is locally algebraizable at the origin,then all the derivative functions ∂ y ′kψ l ′(y′ ), where 1 ≤ k, l ≤ m, are realalgebraic functions of y ′ . Equivalently, every second derivative ∂y 2 k y lϕ j (y) isan algebraic function of the collection of first derivatives ∂ y1 ϕ j , . . .,∂ ym ϕ j .By contraposition, every real analytic strong tube M ∈ Tn d for whichone of the derivative functions ∂ y ′kψ l ′ is not real algebraic is not locally algebraizable.We will argue in §8 that this is generically the case in the senseof Baire. It is however natural to look for explicit examples of nonalgebraizablereal analytic submanifolds in C n . Since the real parts of the vectorfields ∂ z1 , . . .,∂ zm , ∂ w1 , . . ., ∂ wd are infinitesimal CR automorphisms of everytube v = ϕ(y), we must provide some sufficient conditions insuringthat the dimension of the Lie algebra of such a tube is exactly n. We shallestablish in §§7-8 below:Corollary 1.2. The tube hypersurface M χ1 ,...,χ n−1in C n of equation v =∑ n−1k=1 [ε kyk 2+y6 k +y9 k y 1 · · ·y k−1 +y n+8kχ k (y 1 , . . .,y n−1 )], where χ 1 , . . .,χ n−1are arbitrary real analytic functions, belongs to the class Tn 1 of strong tubes.Two such tubes M χ1 ,...,χ n−1and M bχ1 ,...,bχ n−1are biholomorphically equivalentif and only if χ j = ̂χ j for every j. Furthermore, for a generic choice inχ 1 , . . .,χ n−1 in the sense of Baire (to be precised in §8), M χ1 ,...,χ n−1is notlocally algebraizable at the origin.Here we annihilate some Taylor coefficients in ϕ and keep some othersto be nonzero to insure that M χ is a strong tube. Furthermore, the termsy 9 k y 1 · · ·y k−1 insure that the M χ are pairwise not biholomorphically equivalent.Using a classical direct algorithm (cf. [Bs1991], [St1991]), or the Lietheory of symmetries of differential equations, combined with Theorem 1.1we may provide some other explicit strong tubes which are not locally algebraizable(see §§7-8 for the proof):Corollary 1.3. The following five explicit tubes belong to T2 1 and are notlocally algebraizable at the origin : v = sin(y 2 ), v = tan(y 2 ), v = e ey−1 −1,v = sinh(y 2 ) and v = tanh(y 2 ).In these five examples, the algebraic independence in ∂ y ϕ and in ∂ 2 yy ϕis clear; however, checking that each hypersurface is indeed a strong tuberequires some formal computations, see §7. One may also check by a directcomputation that in a neighborhood of every point p = (z p , w p ) withz p ≠ 0, the hypersurface M HJY of global equation Im w = e |z|2 − 1 is astrong tube (see §7.5). Since it can be represented in a neighborhood of punder the tube form v ′ = e |zp|2 (e y′ −1) − 1 by means of the local change of