Linda Preiss Rothschild

Hypoelliptic differential operators and nilpotent groups

2. Sufficient condi t ions for hypoe l l ip t i c i ty . . . . . . . . . . . . . . . . . . . . . 251 8. Graded a n d free Lie a lgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4. H a r m o n i c analysis on iV a n d the p roof of T h eo rem 2 . . . . . . . . . . . . . . . 257 5. Di la t ions a n d h o m o g e n e i t y on groups . . . . . . . . . . . . . . . . . . . . . . 261 6. Smoo th ly va ry ing families of f u n d a m e n t a l solut ions . . . . . . . . . . . . . . . . 265

Real Submanifolds in Complex Space and Their Mappings (PMS-47)

PrefaceCh. IHypersurfaces and Generic Submanifolds in C[superscript N]3Ch. IIAbstract and Embedded CR Structures35Ch. IIIVector Fields: Commutators, Orbits, and Homogeneity62Ch. IVCoordinates for Generic Submanifolds94Ch. VRings of Power Series and Polynomial Equations119Ch. VIGeometry of Analytic Discs156Ch. VIIBoundary Values of Holomorphic Functions in Wedges184Ch. VIIIHolomorphic Extension of CR Functions205Ch. IXHolomorphic Extension of Mappings of Hypersurfaces241Ch. XSegre Sets281Ch. XINondegeneracy Conditions for Manifolds315Ch. XIIHolomorphic Mappings of Submanifolds349Ch. XIIIMappings of Real-algebraic Subvarieties379References390Index401

Algebraicity of holomorphic mappings between real algebraic sets in Cn

0. Introduction 1. Holomorphic nondegenera~y of real-analytic manifolds 1.1. Preliminaries on real submanifolds of C N 1.2. Holomorphic nondegeneracy and its propagation 1.3. The Levi number and essential finiteness 1.4. Holomorphic nondegeneracy of real algebraic sets 2. The Segre sets of a real~analytic CR submanifold 2.1. Complexification of M, involution, and projections 2.2. Definition of the Segre sets of M at p0 2.3. Homogeneous submanifolds of CR dimension 1 2.4. Homogeneous submanifolds of arbitrary CR dimension 2.5. Proof of Theorem 2.2.1 3. Algebraic properties of holomorphic mappings between real algebraic sets 3.1. A generalization of Theorems 1 and 4 3.2. Propagation of algebralcity 3.3. Proof of Theorem 3.1.2 3.4. Proof of Theorem 3.1.8 3.5. An example 3.6. Proofs of Theorems 1 through 4

Local geometric properties of real submanifolds in complex space

We survey some recent results on local geometric properties of real submanifolds of complex space. Our main focus is on the structure and properties of mappings between such submanifolds. We relate these results to the classification of real submanifolds under biholomorphic, algebraic, or formal transformations. Examples and open problems in this context are also mentioned.

CR structures with group action and extendability of CR functions

Part I presents results on local embedding of CR structures. We consider an abstract CR manifold whose structure is invariant under a transversal Lie group action. We show that such a manifold can always be locally embedded in complex space as a generic submanifold. The proof is based on selection of canonical coordinates and repeated use of the Newlander-Nirenberg theorem (13). When the Lie group is abelian the embedding can be given a particularly simple form. Let l~ 1 be the codimension of our submanifold (called M throughout the paper); it is then convenient to denote by n + I the dimension of the ambient complex space and by zl,..., z,, w 1 .... , w t the complex coordinates; we shall systematically write

Factoring Polynomials Over Algebraic Number Fields

A method for factoring polynomials whose coefficients are in an algebraic number field is presented. This method is a natural extension of the usual Henselian technique for factoring polynomials with integral coefficients. In addition to working in any number field, our algorithm has the advantage of factoring nonmonic polynomials without inordinately increasing the amount of work, essentially by allowing denominators in the coefficients of the polynomial and its factors. We have not yet written a computer program to implement this algorithm, but we give enough information, including examples, to enable the assiduous reader to program his own version. He will need a lot of enthusiasm and the ability to handle polynomials in two variables with large integer coefficients or with coefficients taken modulo p. The examples given in Section 11 and the notes and comments given in Section 12 should be helpful. Knuth [2] contains most of the information that is necessary for coding this algorithm. Narkiewicz [4] is an excellent reference for algebraic number theory. All recent work on polynomial factoring owes a great debt to Zassenhaus [5], in which the Henselian technique for factorization was suggested, and to Berlekamp (see [1] and its bibliography). 2. NOTATION

Convergence and finite determination of formal CR mappings

In this paper, we study the convergence and finite determination of formal holomorphic mappings of (C , 0) taking one real submanifold into another. By a formal (holomorphic) mapping H : (C , 0) → (C , 0), we mean an N -vector H = (H1, . . . , HN ), where each Hj is a formal power series in N indeterminates with no constant term. If M and M ′ are real smooth submanifolds through 0 in C defined near the origin by ρ(Z, Z) = 0 and ρ′(Z, Z) = 0 respectively, where ρ and ρ′ are vector valued smooth defining functions, then we say that a formal mapping H : (C , 0)→ (C , 0) sends M into M ′ if the vector valued power series ρ′(H(Z), H(Z)) is a (matrix) multiple of ρ(Z, Z). For real-analytic hypersurfaces, we shall prove the following. Theorem 1. Let M and M ′ be real-analytic hypersurfaces through the origin in C , N ≥ 2. Assume that neither M nor M ′ contains a nontrivial holomorphic subvariety through 0. Then any formal mapping H : (C , 0)→ (C , 0) sending M into M ′ is convergent. The condition that M ′ above does not contain a nontrivial holomorphic subvariety is necessary (see Remark 2.3). As a corollary we obtain the following characterization.

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂN are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂN are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.

Transversality of CR mappings

We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of CN into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism.

A general reflection principle in C2

Abstract A real analytic hypersurface M through 0 in Cn is said to have the reflection property if any holomorphic mapping defined on one side of M, not totally degenerate at 0, and mapping M into another real analytic hypersurface in Cn, extends holomorphically to a full neighborhood of 0 in Cn. The main result of this paper is that a real analytic hypersurface in C2 has the reflection property if and only if it is not Levi flat.