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

The last statements of Corollaries 1.2 and 1.6 are a direct consequence ofthe following lemma.Lemma 8.1. The set of holomorphic functions ϕ ∈ O k (∆ n ) such that thereexists a polynomial P such that(8.1) P(z, j k ϕ(z)) ≡ 0,is of first category, namely it can be represented as the countable union ofnowhere dense closed subsets. Conversely, the set of functions ϕ ∈ O k (∆ n )such that there is no algebraic dependence relation like (8.1) is generic inthe sense of Baire, namely it can be represented as the countable intersectionof everywhere dense open subsets.Proof. Let N ∈ N. Consider the set F N of functions ϕ such that there existsa polynomial of degree N satisfying (8.1). It suffices to show that F Nis closed and that its complement is everywhere dense. Suppose that a sequence(ϕ (m) ) m∈N converges to ϕ ∈ O k (∆ n ). Let the zero-set of a degreeN polynomial P (m)N(z, J k) contain the graph of the k-jet of ϕ (m) . The coefficientsof P (m)Nbelong to a certain complex projective space P A(C), wherethe integer A = A(n, k) is independent of m. By compactness of P A (C),passing to a subsequence if necessary, the P (m)Nconverge to a nonzero polynomialP N . By continuity, P N (z, j k ϕ(z)) = 0 for all z ∈ O(∆ n ). We claimthat the complement of the union of the F N is dense in O k (∆ n ). Indeed,let ϕ(z) be such that there exists a degree N polynomial P satisfying (8.1).Fix z 0 ∈ ∆ n having rational real and imaginary parts. Then the complexnumbers z 0 , ∂z αϕ(z 0), |α| ≤ k, are algebraically dependent. By a Cantorianargument, there exists complex numbers χ α 0 arbitrarily close to ∂α z ϕ(z 0)such that z 0 , χ α 0 are algebraically independent. Let χ(z) be a polynomialwith ∂z α (z 0 ) = χ α 0 − ∂t α ϕ(z 0 ). We can choose χ to be arbitrarily close tozero in the C k (∆ n ) norm. Then the function ϕ(z) + χ(z) is not Nash algebraic.REFERENCES141[Ar1974][BER1999][BER2000]ARNOLD, V.I.: Équations différentielles ordinaires. Champs de vecteurs,groupes à un paramètre, difféomorphismes, flots, systèmes linéaires, stabilitédes positions d’équilibre, théorie des oscillations, équations différentiellessur les variétés. Traduit du russe par Djilali Embarek, Éditions Mir, Moscou,1974. 267pp.BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Rational dependenceof smooth and analytic CR mappings on their jets. Math. Ann. 315 (1999),205–249.BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Local geometric propertiesof real submanifolds in complex space, Bull. Amer. Math. Soc. 37(2000), no.3, 309–336.