Preface |
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xi | |
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Chapter I. Hypersurfaces and Generic Submanifolds in C^(N) |
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3 | (32) |
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1.1. Real Hypersurfaces in C^(N) |
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3 | (3) |
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1.2. Holomorphic and Antiholomorphic Vectors |
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6 | (3) |
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1.3. CR, Totally Real, and Generic Submanifolds |
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9 | (5) |
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1.4. CR Vector Fields and CR Functions |
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14 | (3) |
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1.5. Finite Type and Minimality Conditions |
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17 | (4) |
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1.6. Coordinate Representations for CR Vector Fields |
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21 | (5) |
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1.7. Holomorphic Extension of CR Functions |
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26 | (4) |
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1.8. Local Coordinates for CR Manifolds |
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30 | (5) |
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Chapter II. Abstract and Embedded CR Structures |
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35 | (27) |
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2.1. Formally Integrable Structures on Manifolds |
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35 | (5) |
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2.2. Levi Form and Levi Map of an Abstract CR Manifold |
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40 | (9) |
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49 | (3) |
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2.4. Approximation Theorem for Continuous Solutions |
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52 | (5) |
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2.5. Further Approximation Results |
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57 | (5) |
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Chapter III. Vector Fields: Commutators, Orbits, and Homogeneity |
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62 | (32) |
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62 | (6) |
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68 | (5) |
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3.3. Local Orbits of Real-analytic Vector Fields |
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73 | (1) |
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3.4. Canonical Forms for Real Vector Fields of Finite Type |
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73 | (14) |
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3.5. Canonical Forms for Real Vector Fields of Infinite Type |
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87 | (4) |
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3.6. Weighted Homogeneous Real Vector Fields |
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91 | (3) |
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Chapter IV. Coordinates for Generic Submanifolds |
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94 | (25) |
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4.1. CR Orbits, Minimality, and Finite Type |
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94 | (1) |
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4.2. Normal Coordinates for Generic Submanifolds |
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95 | (6) |
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4.3. Canonical Coordinates for Generic Submanifolds |
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101 | (7) |
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4.4. Weighted Homogeneous Generic Submanifolds |
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108 | (4) |
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4.5. Normal Canonical Coordinates |
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112 | (7) |
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Chapter V. Rings of Power Series and Polynomial Equations |
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119 | (37) |
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5.1. Finite Codimensional Ideals of Power Series Rings |
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119 | (9) |
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5.2. Analytic Subvarieties |
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128 | (4) |
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5.3. Weierstrass Preparation Theorem and Consequences |
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132 | (7) |
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5.4. Algebraic Functions, Manifolds, and Varieties |
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139 | (6) |
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5.5. Roots of Polynomial Equations with Holomorphic Coefficients |
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145 | (11) |
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Chapter VI. Geometry of Analytic Discs |
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156 | (28) |
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6.1. Hilbert and Poisson Transforms on the Unit Circle |
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156 | (6) |
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6.2. Analytic Discs Attached to a Generic Submanifold |
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162 | (4) |
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6.3. Submanifolds of a Banach Space |
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166 | (10) |
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6.4. Mappings of the Banach Space C^(1.Alpha) |
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176 | (2) |
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6.5. Banach Submanifolds of Analytic Discs |
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178 | (6) |
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Chapter VII. Boundary Values of Holomorphic Functions in Wedges |
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184 | (21) |
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7.1. Wedges with Generic Edges in C^(N) |
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184 | (1) |
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7.2. Holomorphic Functions of Slow Growth in Wedges |
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185 | (7) |
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7.3. Continuity of Boundary Values |
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192 | (4) |
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7.4. Uniqueness of Boundary Values |
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196 | (6) |
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7.5. Additional Smoothness up to the Edge |
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202 | (2) |
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7.6. Further Results and an "Edge-of-the-Wedge" Theorem |
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204 | (1) |
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Chapter VIII. Holomorphic Extension of CR Functions |
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205 | (36) |
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8.1. Criteria for Wedge Extendability of CR Functions |
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205 | (1) |
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8.2. Sufficient Conditions for Filling Open Sets with Discs |
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206 | (6) |
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8.3. Tangent Space to the Manifold of Discs |
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212 | (6) |
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8.4. Defect of an Analytic Disc Attached to a Manifold |
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218 | (6) |
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8.5. Ranks of the Evaluation and Derivative Maps |
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224 | (6) |
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8.6. Minimality and Extension of CR Functions |
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230 | (1) |
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8.7. Necessity of Minimality for Holomorphic Extension to a Wedge |
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231 | (7) |
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8.8. Further Results on Wedge Extendability of CR Functions |
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238 | (3) |
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Chapter IX. Holomorphic Extension of Mappings of Hypersurfaces |
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241 | (40) |
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9.1. Reflection Principle in the Complex Plane |
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242 | (1) |
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9.2. Reflection Principle: Preliminaries |
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243 | (3) |
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9.3. Reflection Principle for Levi Nondegenerate Hypersurfaces |
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246 | (2) |
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9.4. Essential Finiteness for Real-analytic Hypersurfaces |
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248 | (4) |
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9.5. Formal Power Series of CR Mappings |
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252 | (3) |
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9.6. Reflection Principle for Essentially Finite Hypersurfaces |
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255 | (2) |
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9.7. Polynomial Equations for Components of a Mapping |
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257 | (2) |
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9.8. End of Proof of the Reflection Principle |
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259 | (6) |
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9.9. Reflection Principle for CR Mappings |
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265 | (5) |
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9.10. Reflection Principle for Bounded Domains |
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270 | (7) |
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9.11. Further Results on the Reflection Principle |
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277 | (4) |
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281 | (34) |
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10.1. Complexification of a Generic Real-analytic Submanifold |
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281 | (2) |
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10.2. Definition of the Segre Manifolds and Segre Sets |
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283 | (6) |
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10.3. Examples of Segre Sets and Segre Manifolds |
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289 | (4) |
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10.4. Basic Properties of the Segre Sets |
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293 | (7) |
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10.5. Segre Sets, CR Orbits, and Minimality |
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300 | (5) |
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10.6. Homogeneous Submanifolds of CR Dimension One |
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305 | (7) |
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10.7. Proof of Theorem 10.5.2 |
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312 | (3) |
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Chapter XI. Nondegeneracy Conditions for Manifolds |
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315 | (34) |
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11.1. Finite Nondegeneracy of Abstract CR Manifolds |
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315 | (4) |
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11.2. Finite Nondegeneracy of Generic Submanifolds of C^(N) |
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319 | (3) |
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11.3. Holomorphic Nondegeneracy |
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322 | (3) |
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11.4. Essential Finiteness for Real-analytic Submanifolds |
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325 | (4) |
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11.5. Comparison of Nondegeneracy Conditions |
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329 | (6) |
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11.6. Compact Real-analytic Generic Submanifolds |
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335 | (1) |
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11.7. Nondegeneracy for Smooth Generic Submanifolds |
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336 | (6) |
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11.8. Essential Finiteness of Smooth Generic Submanifolds |
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342 | (7) |
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Chapter XII. Holomorphic Mappings of Submanifolds |
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349 | (30) |
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12.1. Jet Spaces and Jets of Holomorphic Mappings |
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349 | (3) |
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12.2. Basic Identity for Holomorphic Mappings |
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352 | (6) |
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12.3. Determination of Holomorphic Mappings by Finite Jets |
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358 | (3) |
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12.4. Infinitesimal CR Automorphisms |
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361 | (5) |
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12.5. Finite Dimensionality of Infinitesimal CR Automorphisms |
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366 | (4) |
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12.6. Iterations of the Basic Identity |
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370 | (3) |
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12.7. Analytic Dependence of Mappings on Jets |
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373 | (6) |
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Chapter XIII. Mappings of Real-algebraic Subvarieties |
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379 | (11) |
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13.1. Mappings between Generic Real-algebraic Submanifolds |
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379 | (4) |
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13.2. Some Necessary Conditions for Algebraicity of Mappings |
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383 | (4) |
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13.3. Mappings of Real-algebraic Subvarieties |
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387 | (3) |
References |
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390 | (11) |
Index |
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401 | |