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

98χ k = ̂χ k for k = 1, . . .,n − 1. Furthermore, for a generic choice of a(n − 1)-tuple of real analytic functions (χ 1 , . . .,χ n−1 ) in the sense of Baire(to be precised in §8), M χ1 ,...,χ n−1is not locally algebraizable at the origin.Finally, by computing generators of the Lie algebra of local infinitesimalCR automorphisms of some explicit examples, we obtain:Corollary 1.7. The following seven explicit examples of hypersurfaces inC 2 are strongly rigid and are not locally algebraizable at the origin : v =z¯z + z 2¯z 2 sin(z + ¯z), v = z¯z + z 2¯z 2 exp(z + ¯z), v = z¯z + z 2¯z 2 cos(z + ¯z),v = z¯z+z 2¯z 2 tan(z+¯z), v = z¯z+z 2¯z 2 sinh(z+¯z), v = z¯z+z 2¯z 2 cosh(z+¯z)and v = z¯z + z 2¯z 2 tanh(z + ¯z).1.3. Content of the paper. To prove Theorem 1.1 we consider an algebraicequivalent M ′ of M. The main technical part of the proof consists in showingthat an arbitrary real algebraic element M ′ of Tn d can be straightened insome local complex algebraic coordinates t ′ ∈ C n in order that its infinitesimalCR automorphisms are the real parts of n holomorphic vector fields ofthe form X i ′ = c′ i (t′ i ) ∂ t ′ , i = 1, . . .,n, where the variables are separated andithe functions c ′ i (t′ i ) are algebraic. For this, we need to show that the automorphismgroup of a minimal finitely nondegenerate real algebraic genericsubmanifold in C n is a local real algebraic Lie group, a notion defined in§2.3. A large part of this article (§§4, 5, 6) is devoted to provide an explicitrepresentation formula for the local biholomorphic self-transformations ofa minimal finitely nondegenerate generic submanifold, see especially Theorem2.1 and Theorem 4.1. Finally, using the specific simplified form ofthe vector fields X i ′ and assuming that there exists a biholomorphic equivalenceΦ : M → M ′ satisfying Φ ∗ (∂ ti ) = c ′ i (t′ i ) ∂ t ′ , we show by elementaryicomputations that all the first order derivatives of the mapping ψ ′ (y ′ ) mustbe algebraic. We follow a similar strategy for the proofs of Theorems 1.4and 1.5. Finally, in §§7-8, we provide the proofs of Corollaries 1.2, 1.3, 1.6and 1.7.1.4. Acknowledgment. We acknowledge interesting discussions withMichel Petitot François Boulier at the University of Lille 1.§2. PRELIMINARIESWe recall in this section the basic properties of the objects we will dealwith.2.1. Nash algebraic functions and manifolds. In this subsection, let K =R or C. Let (x 1 , . . .,x n ) denote coordinates over K n . Throughout thearticle, we shall use the norm |x| := max(|x 1 |, . . .,|x n |) for x ∈ K n .Let K be an open polydisc centered at the origin in K n , namely K =