A Primer of Algebraic Geometry: Constructive Computational Methods

Constructive Computational Methods

By: Huishi Li, Freddy Van Oystaeyen

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Hardcover | 1 January 2000 | Edition Number 1

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Hardcover
394 Pages

Dimensions(cm)
23.39 x 15.6 x 2.24

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Li (mathematics, Bilkent U. Turkey) and Oystaeyen (mathematics, U. of Antwerp/UIA, Belgium) emphasize the new Groebner basis method that combines ordered ideal structure theory in polynomial rings with the Weierstrass theory of elliptic curves. Designed both as a text for high level undergraduate or first-year graduate students and as a reference for pure and applied mathematicians, this book covers a range of material, including: polynomials, affine space, radical ideals, rational functions, irreducible algebraic sets, the Riemann-Roch theorem, elliptic functions, and the Zariski topology.

Industry Reviews

"This book can be used as a first course in algebraic geometry for students and researchers who are not primarily pure mathematicians. It is also useful for applications in computer algebra, robotics and computational geometry and mathematical methods in technology. I wish to recommend this well-written book to anyone interested in applied algebraic geometry." - EMS Newsletter, June 2003

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Preface	p. iii
Notation	p. xiii
Affine Algebraic Sets and the Nullstellensatz	p. 1
Polynomials and Affine Space	p. 1
Affine Algebraic Sets	p. 9
Parametrizations of Algebraic Sets	p. 11
Ideals for Algebraic Sets	p. 21
Hilbert's Nullstellensatz	p. 22
Radical Ideals and the Nullstellensatz	p. 27
Polynomial and Rational Functions	p. 33
The Zariski Topology and Irreducible Algebraic Sets	p. 34
Decomposition of an Algebraic Set	p. 40
Polynomial Mappings and Polynomial Functions	p. 43
The Coordinate Ring of an Algebraic Set	p. 48
Affine Change of Coordinates	p. 56
Rational Functions and Local Rings	p. 58
Projective Algebraic Sets	p. 69
Projective Space	p. 71
The Projective Algebra-Geometry Dictionary	p. 78
Homogeneous Coordinate Ring and Function Field	p. 82
Projective Change of Coordinates	p. 85
Dehomogenization and Homogenization of Polynomials	p. 90
Affine-Projective Transfer of Algebraic Sets	p. 95
Multiprojective Space and Segre Product	p. 106
Groebner Basis	p. 111
Monomial Orderings	p. 111
A Division Algorithm in k[x[subscript 1], ..., x[subscript n]]	p. 120
Monomial Ideals and Dickson's Lemma	p. 128
Hilbert Basis Theorem and Groebner Basis	p. 133
Applications to Previous Chapters	p. 139
Dimension of Algebraic Sets	p. 147
The Algebraic Set of a Monomial Ideal	p. 150
The Vector Space k[x[subscript 1], ..., x[subscript n]]/I	p. 155
Hilbert Function and dimV	p. 159
The Dimension of a Projective Algebraic Set	p. 173
Elementary Properties of Dimension	p. 179
Dimension and Algebraic Independence	p. 186
Dimension and Elimination of Variables	p. 192
The Krull Dimension of an Affine k-Algebra	p. 200
The Topological Dimension of an Affine Algebraic Set	p. 205
An Introduction to Local Theory	p. 207
Regular Functions on Quasi-Varieties	p. 209
Morphisms	p. 215
Rational Maps	p. 225
Examples of Rational Varieties	p. 233
An Algorithm Checking the Birationality of Rational Maps	p. 238
Nonsingular Points in Algebraic Sets	p. 240
Curves	p. 253
Nonsingular Curves	p. 253
Intersection of Curves	p. 265
Bezout's Theorem	p. 277
Divisors on Curves	p. 283
The Linear Space L(D)	p. 290
The Riemann-Roch Theorem	p. 297
Elliptic Curves	p. 307
Standard Equation for Nonsingular Cubic Curve	p. 308
Addition on an Elliptic Curve	p. 315
Elliptic Functions and Weierstrass Theory	p. 327
Finiteness Conditions and Field Extensions	p. 343
Modules, Finiteness Conditions	p. 343
Integral Elements	p. 346
Ring-finite Field Extensions	p. 348
Existence of a Primitive Element	p. 350
Localization, Discrete Valuation Rings and Dedekind Domains	p. 355
Localization at a Multiplicative Subset	p. 355
Discrete Valuation Rings	p. 357
Dedekind Domains	p. 361
References	p. 365
Index	p. 367
Table of Contents provided by Syndetics. All Rights Reserved.

A Primer of Algebraic Geometry: Constructive Computational Methods

Constructive Computational Methods

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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