The Basic Theory of Power Series

Overview

Power series techniques are indispensable in many branches of mathematics, in particular in complex and in real analytic geometry, in commutative algebra, in algebraic geometry, in real algebraic geometry. The book covers in a comprehensive way and at an elementary level essentially all the theorems and techniques which are commonly used and needed in any of these branches. In particular it presents R?ckert's complex nullstellensatz, Risler's real nullstellensatz, Tougerons' implicit function theorem, and Artin's...

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Overview

Power series techniques are indispensable in many branches of mathematics, in particular in complex and in real analytic geometry, in commutative algebra, in algebraic geometry, in real algebraic geometry. The book covers in a comprehensive way and at an elementary level essentially all the theorems and techniques which are commonly used and needed in any of these branches. In particular it presents Rückert's complex nullstellensatz, Risler's real nullstellensatz, Tougerons' implicit function theorem, and Artin's approximation theorem, to name a few. Up to now a student of any of the subjects mentioned above usually had to learn about power series within the framework of the vast theory of the subject. The present book opens another path: One gets acquaintance with power series in a direct and elementary way, and then disposes of a good box of tools and examples to penetrate any of the subjects mentioned above, and also some others.

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Editorial Reviews

Booknews
For advanced undergraduate and early graduate mathematics students, explains power series techniques and theorems, and demonstrates their application in a number of branches of mathematics. Most students are exposed to power series only within the context of a specific application. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Meet the Author

Dr. Jésus M. Ruiz ist Professor für Mathematik am Institut für Geometrie und Topologie an der Universität Complutense de Madrid.

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Table of Contents

I Power Series.- 1 Series of Real and Complex Numbers.- 2 Power Series.- 3 Rückert’s and Weierstrass’s Theorems.- II Analytic Rings and Formal Rings.- 1 Mather’s Preparation Theorem.- 2 Noether’s Projection Lemma.- 3 Abhyankar’s and Rückert’s Parametrization.- 4 Nagata’s Jacobian Criteria.- 5 Complexification.- III Normalization.- 1 Integral Closures.- 2 Normalization.- 3 Multiplicity in Dimension 1.- 4 Newton-Puiseux’s Theorem.- IV Nullstellensatze.- 1 Zero Sets and Zero Ideals.- 2 Rückert’s Complex Nullstellensatz.- 3 The Homomorphism Theorem.- 4 Risler’s Real Nullstellensatz.- 5 Hilbert’s 17th Problem.- V Approximation Theory.- 1 Tougeron’s Implicit Functions Theorem.- 2 Equivalence of Power Series.- 3 M. Artin’s Approximation Theorem.- 4 Formal Completion of Analytic Rings.- 5 Nash Rings.- VI Local Algebraic Rings.- 1 Local Algebraic Rings.- 2 Chevalley’s Theorem.- 3 Zariski’s Main Theorem.- 4 Normalization and Completion.- 5 Efroymson’s Theorem.- Bibliographical Note.

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