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

186coordinates (z, w) = (x + iy, u + iv) vanishing at the point p, the definingequation is of the so-called rigid form u = ϕ(x, y) with the variable vmissing, or even of the so-called (simpler) tube form u = ϕ(x), with thetwo variables y and v missing, see [11] which showed recently a renewedinterest, in CR geometry, for explicit characterizations of sphericality. Butin general, a real analytic hypersurface M ⊂ C 2 is represented at p by a realequation u = ϕ(x, y, v) whose graphing function ϕ depends entirely arbitrarilyupon v also, and then apparently, the characterization of sphericalityis still unknown.On the other hand, in the studies [12, 13, 14, Me2005a, Me2005b] devotedto the CR reflection principle, it was emphasized that all the adequateinvariants of CR mappings between CR manifolds: Pair of Segre foliations,Segre chains, Complexified CR orbits, Jets of complexified Segre varietes,Rigidity of formal CR mappings, Nondegeneracy conditions, CR-reflectionfunction 5 , can be viewed correctly only when M is represented by a socalledcomplex defining equation of the form:w = Θ ( z, z, w ) ,where the function Θ ∈ C { z, z, w } , vanishing at the origin, is the uniquew+wfunction obtained by solving with respect to w the equation: =2ϕ ( z+z, z−z,) w−w2 2i 2i ; then the fact that ϕ was real is reflected, in terms of thisnew function Θ(z, z, w), by the constraint that, together with its complexconjugate Θ ( z, z, w ) , it satisfies the functional equation 6 :w ≡ Θ ( z, z, Θ(z, z, w) ) .Accordingly, the author suspected since a few years — cf. the Open Question2.35 in [19] — that sphericality of M at p should and could be expressedadequately in terms of Θ. The classical assumption that M be Levinondegenerate at the point p (see e.g. [11]) — which is the origin of ourpresent system of coordinates (z, w) — may then be expressed here (cf.[Me2005a, Me2005b]) by requiring that Θ z Θ zw −Θ w Θ zz does not vanish atthe origin. In particular, this guarantees that the following explicit rationalexpression whose numerator is a polynomial in the fourth-order jet J 4 z,z,w Θ,is well defined and analytic in some sufficiently small neighborhood of the5 For a presentation of these concepts, the reader is referred to the extensive introductionsof [14, Me2005b] and also to [19] for more about why dealing only with complexdefining equations is natural and unavoidable when one wants to insert CR geometry in thewider universe of completely integrable systems of real or complex analytic partial differentialequations.6 More will be said shortly in Section 2 below.