Rational dependence of smooth and analytic CR mappings on their jets

Baouendi, M. S.; Ebenfelt, Peter; Rothschild, Linda Preiss

doi:10.1007/s002080050365

Cited by 52 publications

(109 citation statements)

References 23 publications

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“…Let V be the subfield of FN+U defined by (4.6). We also recall that for the positive integer ZQ mentioned in Remark 1, Vi Q is the smallest field contained in TN+U an d containing C and the family (Oi^OPjiuityi^ctflKlo)' Let X € C. Since, by Proposition 4.1, x is algebraic over V and, according to Remark 1, also over Vi Qi we obtain the existence of a positive integer ko and a family (bj(u, 0))o<j</co-i C P/ 0 such that the following identity (4.11) ( holds, with the additional non-degeneracy condition (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) …”

Section: Is the Complexification Of M As Defined By (32)mentioning

confidence: 94%

“…(We also refer the reader to the bibliography given in [6] for further information.) More recently, Baouendi, Ebenfelt and Rothschild proved the convergence of formal equivalences between minimal finitely nondegenerate real analytic generic submanifolds [4,5], as well as between minimal essentially finite ones 1 [6] (other situations are also treated in [5,6]). The conditions of finite nondegeneracy and essential finiteness are closely related to the notion of holomorphic nondegeneracy introduced by Stanton [24].…”

Section: Pu{z) F(z)) = A(z Z) • P(z Z)mentioning

confidence: 99%

“…By applying the vector fields Cj, j = 1,..., A/", to (4.9) and Cramer's rule, one obtains the following known lemma (cf. [5,12]). The following lemma contains two known and easy facts.…”

Section: Is the Complexification Of M As Defined By (32)mentioning

confidence: 99%

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On the convergence of formal mappings

Mir¹

2002

Communications in Analysis and Geometry

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Let f : (M, p) → (M ′ , p ′ ) be a formal (holomorphic) nondegenerate map, i.e. with formal holomorphic Jacobian J f not identically vanishing, between two germs of real analytic generic submanifolds in C n , n ≥ 2, p ′ = f (p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f . From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds into real algebraic holomorphically nondegenerate ones, as well as related results on partial convergence of such maps. For the proofs, we establish a principle of analyticity for formal CR power series. This principle can be used to reobtain the convergence of formal mappings of real analytic CR manifolds under a standard nondegeneracy condition.

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Section: Is the Complexification Of M As Defined By (32)mentioning

confidence: 94%

Section: Pu{z) F(z)) = A(z Z) • P(z Z)mentioning

confidence: 99%

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On the convergence of formal mappings

Mir¹

2002

Communications in Analysis and Geometry

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“…In the real-analytic case, results on jet parametrization were obtained by the first two authors jointly with Ebenfelt [BER97,BER99a] and by the fourth author [Z97]. The partial converse mentioned above is stated in both global and local cases in Theorems 2.4 and 2.10.…”

mentioning

confidence: 98%

Lie group structures on groups of diffeomorphisms and applications to CR manifolds

Baouendi¹,

Rothschild²,

Winkelmann³

et al. 2004

Annales De L’institut Fourier

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“…In that paper, the image of the mapping v j was called the jth Segre set, and the existence of an integer k 0 such that (3) holds was established. Also, for M merely smooth, the existence of the integer k 0 such that (i) implies (ii) in Theorem 1.1 can be deduced from [BER96] by again using the special canonical coordinates mentioned above (see [BER99b] and [BER99c]). However, the results in Theorem 1.1, even in the case where M is real-analytic, are sharper than those mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of the Segre varieties of a real submanifold in complex space

2002

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For a smooth (or formal) generic submanifold M M of real codimension d d in complex space C N \mathbb {C}^N with 0 ∈ M 0\in M , we introduce the notion of a formal Segre variety mapping γ : ( C N × C N − d , 0 ) → ( C N , 0 ) \gamma : (\mathbb {C}^N\times \mathbb {C} ^{N-d},0)\to (\mathbb {C}^N,0) and its iterated Segre mappings at 0 0 , v j : ( C ( N − d ) j , 0 ) → ( C N , 0 ) v^j:(\mathbb {C}^{(N-d)j},0) \to (\mathbb {C}^N,0) , j ≥ 1 j\ge 1 . The Segre variety mapping γ \gamma extends the notion of Segre varieties of a real-analytic generic submanifold to the setting of smooth (or formal) submanifolds. One of the main results in this paper is that M M is of finite type (in the sense of Kohn and Bloom–Graham) at 0 0 if and only if there exists k 0 ≤ d + 1 k_0\le d+1 such that the (generic) rank of v k 0 v^{k_0} is N N . More generally, we prove that v k 0 v^{k_0} parameterizes the local CR orbit of M M at 0 0 .

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Rational dependence of smooth and analytic CR mappings on their jets

Cited by 52 publications

References 23 publications

On the convergence of formal mappings

On the convergence of formal mappings

Lie group structures on groups of diffeomorphisms and applications to CR manifolds

Dynamics of the Segre varieties of a real submanifold in complex space

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