Book Description
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of
Ideals, Varieties and
Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.
Customer Reviews:
Symbolic computation.......2003-08-29
This book explains and illustrates the algorithms used by symbolic math packages such as Mathematica, Maple, CoCoA, MatLab, MuPAD,... to solve problems involving polynomials in many variables, and along the way teaches the elements of real algebraic geometry-- most mathematics texts concentrate on the complex-variable version. It is not just for undergraduates; electrical engineers, for instance, should see it. Lots of pictures!
Easiest introduction to Algebraic Geometry.......2003-04-23
This is the easiest introduction to algebraic geometry and commutative algebra, the authors had done a great job in writing a book that assume very little from the readers. To learn some algebraic geometry, you can either start with this book, or you can spend a year to read a lot of background materials in algebra and then go to a Graduate Text like Harris' book. Of course, if you want to be an expert in algebra, you eventually need a lot of background, what this book can help you is to offer you a quick start, much quicker than you would ever imagine.
Straightforward and lucidly written.......2002-04-09
Having just finished using this text in the course of an undergraduate seminar, I can attest to the fact that the authors' style is outstanding - they are able to synthesize an enormous amount of material in this volume and present it in a manner that is highly accessible to almost all students of mathematics. The presentation of important theorems (for example, Hilbert's Nullstellensatz and Basis Theorem) along with just the right amount of copncrete examples makes for a book of superb quality. All-around, I highly recommend this volume to anyone who has an interest in learning about Algebraic Geometry.
Good book.......2001-05-27
I don't have the second edition of this book but did read the first, and the authors do a fine job of introducing the reader to the computational side of algebraic geometry. I will forego a chapter by chapter review therefore, but no doubt the second edition (which I do not own) is as well-written as the first. I would recommend it to anyone interested in the many applications of algebraic geometry and to those who need to understand how to compute things in algebraic geometry. The good thing about this book is that it gives a concrete flavor to a highly abstract subject. Algebraic geometry, through its applications to coding theory, cryptography, and computer graphics, is fast becoming the subject to learn. It is no longer just an esoteric, high-brow subject but one that is taking on major importance in the information age. Even without applications though it is a fascinating subject, and readers will get a taste of this in this book.
The best book on the topic.......2001-01-26
I learned the basics of Groebner bases from this book and its the best introductory book on this topic. Authors have explained all concepts with the help of examples which makes it readable for people from other fields also. It also talks about applications of Groebner bases to other fields. The book gives lot of exercises which help in understanding the contents more. I recommend that if you wish to learn Algebraic Geometry and Groebner bases then this is the book to start with.
Book Description
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?
The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.
The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.
In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes: A significantly updated section on Maple in Appendix C; Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3.
From the 2nd edition: "I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." The American Mathematical Monthly
Book Description
Recent advances in computing and algorithms make it easier to do many classical problems in algebra. Suitable for graduate students, this book brings advanced algebra to life with many examples. The first three chapters provide an introduction to commutative algebra and connections to geometry. The remainder of the book focuses on three active areas of contemporary algebra: homological algebra; algebraic combinatorics and algebraic topology; and algebraic geometry.
Download Description
The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Recent advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens wonderful new vistas and allows us to pose, study and solve problems that were previously out of reach. Suitable for graduate students, the objective of this book is to bring advanced algebra to life with lots of examples. The first chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book focuses on three active areas of contemporary algebra: Homological Algebra (the snake lemma, long exact sequence inhomology, functors and derived functors (Tor and Ext), and double complexes); Algebraic Combinatorics and Algebraic Topology (simplicial complexes and simplicial homology, Stanley-Reisner rings, upper bound theorem and polytopes); and Algebraic Geometry (points and curves in projective space, Riemann-Roch, Cech cohomology, regularity).
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Polytopes: Abstract, Convex and Computational (NATO Science Series C: (closed))
Manufacturer: Kluwer Academic
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ASIN: 0792330161 |
Book Description
The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject.
The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex.
With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes.
For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.
Book Description
This book is the natural continuation of
Computational Commutative Algebra 1 with some twists.
The main part of this book is a breathtaking
passeggiata through the computational domains of graded rings and modules and their Hilbert functions. Besides Gröbner bases, we encounter Hilbert bases, border bases, SAGBI bases, and even SuperG bases.
The tutorials traverse areas ranging from algebraic geometry and combinatorics to photogrammetry, magic squares, coding theory, statistics, and automatic theorem proving. Whereas in the first volume gardening and chess playing were not treated, in this volume they are.
This is a book for learning, teaching, reading, and most of all, enjoying the topic at hand. The theories it describes can be applied to anything from children's toys to oil production.
Book Description
This volume is a collection of refereed expository and research articles in discrete and computational geometry written by leaders in the field. Articles are based on invited talks presented at the AMS-IMS-SIAM Summer Research Conference, "Discrete and Computational Geometry: Ten Years Later", held in 1996 at Mt. Holyoke College (So. Hadley, MA). Topics addressed range from tilings, polyhedra, and arrangements to computational topology and visibility problems. Included are papers on the interaction between real algebraic geometry and discrete and computational geometry, as well as on linear programming and geometric discrepancy theory.
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Algorithmic Number Theory: 5th International Symposium, ANTS-V, Sydney, Australia, July 7-12, 2002. Proceedings (Lecture Notes in Computer Science)
Manufacturer: Springer
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ASIN: 3540438637 |
Book Description
This book constitutes the refereed proceedings of the 5th International Algorithmic Number Theory Symposium, ANTS-V, held in Sydney, Australia, in July 2002.The 34 revised full papers presented together with 5 invited papers have gone through a thorough round of reviewing, selection and revision. The papers are organized in topical sections on number theory, arithmetic geometry, elliptic curves and CM, point counting, cryptography, function fields, discrete logarithms and factoring, Groebner bases, and complexity.
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Algorithmic Number Theory: 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings (Lecture Notes in Computer Science)
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback
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ASIN: 3540221565 |
Book Description
This book constitutes the refereed proceedings of the 6th International Algorithmic Number Theory Symposium, ANTS 2004, held in Burlington, VT, USA, in June 2004.
The 30 revised full papers presented together with 3 invited papers were carefully reviewed and selected for inclusion in the book. Among the topics addressed are zeta functions, elliptic curves, hyperelliptic curves, GCD algorithms, number field computations, complexity, primality testing, Weil and Tate pairings, cryptographic algorithms, function field sieve, algebraic function field mapping, quartic fields, cubic number fields, lattices, discrete logarithms, and public key cryptosystems.
Book Description
This book constitutes the refereed proceedings of the 7th International Algorithmic Number Theory Symposium, ANTS 2006, held in Berlin, Germany in July 2006.
The 37 revised full papers presented together with 4 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on algebraic number theory, analytic and elementary number theory, lattices, curves and varieties over fields of characteristic zero, curves over finite fields and applications, and discrete logarithms.
Book Description
This book introduces readers to key ideas and applications of computational algebraic geometry. Beginning with the discovery of Gröbner bases and fueled by the advent of modern computers and the rediscovery of resultants, computational algebraic geometry has grown rapidly in importance. The fact that "crunching equations" is now as easy as "crunching numbers" has had a profound impact in recent years. At the same time, the mathematics used in computational algebraic geometry is unusually elegant and accessible, which makes the subject easy to learn and easy to apply.
This book begins with an introduction to Gröbner bases and resultants, then discusses some of the more recent methods for solving systems of polynomial equations. A sampler of possible applications follows, including computer-aided geometric design, complex information systems, integer programming, and algebraic coding theory. The lectures in the book assume no previous acquaintance with the material.
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