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93Nonalgebraizable real analytic tubes in C n xJoël Merker (with H. Gaussier)Abstract. We give necessary conditions for certain real analytic tube generic submanifoldsin C n to be locally algebraizable. As an application, we exhibit familiesof real analytic non locally algebraizable tube generic submanifolds in C n .During the proof, we show that the local CR automorphism group of a minimal,finitely nondegenerate real algebraic generic submanifold is a real algebraic localLie group. We may state one of the main results as follows. Let M be a real analytichypersurface tube in C n passing through the origin, having a defining equation ofthe form v = ϕ(y), where (z,w) = (x+iy,u+iv) ∈ C n−1 ×C. Assume that M isLevi nondegenerate at the origin and that the real Lie algebra of local infinitesimalCR automorphisms of M is of minimal possible dimension n, i.e. generated by thereal parts of the holomorphic vector fields ∂ z1 ,... ,∂ zn−1 ,∂ w . Then M is locallyalgebraizable only if every second derivative ∂ 2 y k y lϕ is an algebraic function of thecollection of first derivatives ∂ y1 ϕ,...,∂ ym ϕ.Table of contents :§1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.§2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.§3. Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.§4. Local Lie group structure for the CR automorphism group . . . . . . . .17.§5. Minimality and finite nondegeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.§6. Algebraicity of local CR automorphism groups . . . . . . . . . . . . . . . . . . . 22.§7. Description of explicit families of strong tubes in C n . . . . . . . . . . . . . . 28.§8. Analyticity versus algebraicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.Mathematische Zeitschrift 247 (2004), no. 2, 337–383§1. INTRODUCTIONA real analytic submanifold M in C n is called algebraic if it can be representedlocally by the vanishing of a collection of Nash algebraic real analyticfunctions. We say that M is locally algebraizable at one of its points p ifthere exist some local holomorphic coordinates centered at p in which Mis algebraic. For instance, every totally real, real analytic submanifold inC n of dimension k ≤ n is locally biholomorphic to a k-dimensional linearreal plane, hence locally algebraizable. Also, every complex manifoldis locally algebraizable. Although every real analytic submanifold M isclearly locally equivalent to its tangent plane by a real analytic (in general