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Cover: 9783528065256 | The Basic Theory of Power Series | Jesús M. Ruiz | Taschenbuch | x
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The Basic Theory of Power Series
Taschenbuch von Jesús M. Ruiz
Sprache: Englisch

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Beschreibung
The aim of these notes is to cover the basic algebraic tools and results behind the scenes in the foundations of Real and Complex Analytic Geometry. The author has learned the subject through the works of many mathematicians, to all of whom he is indebted. However, as the reader will immediately realize, he was specially influenced by the writings of S.S. Abhyankar and J .-C. Tougeron. In any case, the presentation of all topics is always as elementary as it can possibly be, even at the cost of making some arguments longer. The background formally assumed consists of: 1) Polynomials: roots, factorization, discriminant; real roots, Sturm's Theorem, formally real fields; finite field extensions, Primitive Element Theorem. 2) Ideals and modules: prime and maximal ideals; Nakayama's Lemma; localiza­ tion. 3) Integral dependence: finite ring extensions and going-up. 4) Noetherian rings: primary decomposition, associated primes, Krull's Theorem. 5) Krull dimension: chains of prime ideals, systems of parameters; regular systems of parameters, regular rings. These topics are covered in most texts on Algebra and/or Commutative Algebra. Among them we choose here as general reference the following two: ¿ M. Atiyah, I.G. Macdonald: Introduction to Commutative Algebra, 1969, Addison-Wesley: Massachusetts; quoted [A-McD] . ¿ S. Lang: Algebra, 1965, Addison-Wesley: Massachusetts; quoted [L].
The aim of these notes is to cover the basic algebraic tools and results behind the scenes in the foundations of Real and Complex Analytic Geometry. The author has learned the subject through the works of many mathematicians, to all of whom he is indebted. However, as the reader will immediately realize, he was specially influenced by the writings of S.S. Abhyankar and J .-C. Tougeron. In any case, the presentation of all topics is always as elementary as it can possibly be, even at the cost of making some arguments longer. The background formally assumed consists of: 1) Polynomials: roots, factorization, discriminant; real roots, Sturm's Theorem, formally real fields; finite field extensions, Primitive Element Theorem. 2) Ideals and modules: prime and maximal ideals; Nakayama's Lemma; localiza­ tion. 3) Integral dependence: finite ring extensions and going-up. 4) Noetherian rings: primary decomposition, associated primes, Krull's Theorem. 5) Krull dimension: chains of prime ideals, systems of parameters; regular systems of parameters, regular rings. These topics are covered in most texts on Algebra and/or Commutative Algebra. Among them we choose here as general reference the following two: ¿ M. Atiyah, I.G. Macdonald: Introduction to Commutative Algebra, 1969, Addison-Wesley: Massachusetts; quoted [A-McD] . ¿ S. Lang: Algebra, 1965, Addison-Wesley: Massachusetts; quoted [L].
Über den Autor
Dr. Jésus M. Ruiz ist Professor für Mathematik am Institut für Geometrie und Topologie an der Universität Complutense de Madrid.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 1993
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 148
Reihe: Advanced Lectures in Mathematics
Inhalt: x
134 S.
ISBN-13: 9783528065256
ISBN-10: 3528065257
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Ruiz, Jesús M.
Auflage: 1993
Hersteller: Vieweg & Teubner
Vieweg+Teubner Verlag
Advanced Lectures in Mathematics
Maße: 229 x 162 x 9 mm
Von/Mit: Jesús M. Ruiz
Erscheinungsdatum: 01.01.1993
Gewicht: 0,241 kg
preigu-id: 102304469
Über den Autor
Dr. Jésus M. Ruiz ist Professor für Mathematik am Institut für Geometrie und Topologie an der Universität Complutense de Madrid.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 1993
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 148
Reihe: Advanced Lectures in Mathematics
Inhalt: x
134 S.
ISBN-13: 9783528065256
ISBN-10: 3528065257
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Ruiz, Jesús M.
Auflage: 1993
Hersteller: Vieweg & Teubner
Vieweg+Teubner Verlag
Advanced Lectures in Mathematics
Maße: 229 x 162 x 9 mm
Von/Mit: Jesús M. Ruiz
Erscheinungsdatum: 01.01.1993
Gewicht: 0,241 kg
preigu-id: 102304469
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