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2015 | Buch | 4. Auflage

Geometry, Algebra, and Algorithms Kapitel lesen Erstes Kapitel lesen

Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

verfasst von: David A. Cox, John Little, Donal O’Shea

Verlag: Springer International Publishing

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).

The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.

From the reviews of previous editions:

“…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.”

—Peter Schenzel, zbMATH, 2007

“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

Inhaltsverzeichnis

Frontmatter
Chapter 1. Geometry, Algebra, and Algorithms
Abstract
This chapter will introduce some of the basic themes of the book. The geometry we are interested in concerns affine varieties, which are curves and surfaces (and higher dimensional objects) defined by polynomial equations. To understand affine varieties, we will need some algebra, and in particular, we will need to study ideals in the polynomial ring k[x1, …, xn]. Finally, we will discuss polynomials in one variable to illustrate the role played by algorithms.
David A. Cox, John Little, Donal O’Shea
Chapter 2. Gröbner Bases
Abstract
In Chapter 1, we have seen how the algebra of the polynomial rings k[x1, …, xn] and the geometry of affine algebraic varieties are linked. In this chapter, we will study the method of Gröbner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. The method of Gröbner bases is also used in several powerful computer algebra systems to study specific polynomial ideals that arise in applications. In Chapter 1, we posed many problems concerning the algebra of polynomial ideals and the geometry of affine varieties. In this chapter and the next, we will focus on four of these problems.
David A. Cox, John Little, Donal O’Shea
Chapter 3. Elimination Theory
Abstract
This chapter will study systematic methods for eliminating variables from systems of polynomial equations. The basic strategy of elimination theory will be given in two main theorems: the Elimination Theorem and the Extension Theorem. We will prove these results using Gröbner bases and the classic theory of resultants.
David A. Cox, John Little, Donal O’Shea
Chapter 4. The Algebra–Geometry Dictionary
Abstract
In this chapter, we will explore the correspondence between ideals and varieties. In §§1 and 2, we will prove the Nullstellensatz, a celebrated theorem which identifies exactly which ideals correspond to varieties. This will allow us to construct a “dictionary” between geometry and algebra, whereby any statement about varieties can be translated into a statement about ideals (and conversely). We will pursue this theme in §§3 and 4, where we will define a number of natural algebraic operations on ideals and study their geometric analogues.
David A. Cox, John Little, Donal O’Shea
Chapter 5. Polynomial and Rational Functions on a Variety
Abstract
One of the unifying themes of modern mathematics is that in order to understand any class of mathematical objects, one should also study mappings between those objects, and especially the mappings which preserve some property of interest. For instance, in linear algebra after studying vector spaces, you also studied the properties of linear mappings between vector spaces (mappings that preserve the vector space operations of sum and scalar product).
David A. Cox, John Little, Donal O’Shea
Chapter 6. Robotics and Automatic Geometric Theorem Proving
Abstract
In this chapter we will consider two applications of concepts and techniques from algebraic geometry in areas of computer science. First, continuing a theme introduced in several examples in Chapter 1, we will develop a systematic approach that uses algebraic varieties to describe the space of possible configurations of mechanical linkages such as robot “arms.” We will use this approach to solve the forward and inverse kinematic problems of robotics for certain types of robots.
David A. Cox, John Little, Donal O’Shea
Chapter 7. Invariant Theory of Finite Groups
Abstract
Invariant theory has had a profound effect on the development of algebraic geometry. For example, the Hilbert Basis Theorem and Hilbert Nullstellensatz, which play a central role in the earlier chapters in this book, were proved by Hilbert in the course of his investigations of invariant theory.
David A. Cox, John Little, Donal O’Shea
Chapter 8. Projective Algebraic Geometry
Abstract
So far all of the varieties we have studied have been subsets of affine space kn. In this chapter, we will enlarge kn by adding certain “points at ∞” to create n-dimensional projective space \(\mathbb{P}^{n}(k)\). We will then define projective varieties in \(\mathbb{P}^{n}(k)\) and study the projective version of the algebra–geometry dictionary. The relation between affine and projective varieties will be considered in §4; in §5, we will study elimination theory from a projective point of view. By working in projective space, we will get a much better understanding of the Extension Theorem in Chapter 3 The chapter will end with a discussion of the geometry of quadric hypersurfaces and an introduction to Bezout’s Theorem.
David A. Cox, John Little, Donal O’Shea
Chapter 9. The Dimension of a Variety
Abstract
The most important invariant of a linear subspace of affine space is its dimension. For affine varieties, we have seen numerous examples which have a clearly defined dimension, at least from a naive point of view. In this chapter, we will carefully define the dimension of any affine or projective variety and show how to compute it. We will also show that this notion accords well with what we would expect intuitively. In keeping with our general philosophy, we consider the computational side of dimension theory right from the outset.
David A. Cox, John Little, Donal O’Shea
Chapter 10. Additional Gröbner Basis Algorithms
Abstract
In §10 of Chapter 2 we discussed some criteria designed to identify situations where it is possible to see in advance that an S-polynomial remainder will be zero in Buchberger’s algorithm. Those unnecessary S-polynomial remainder calculations are in fact the main computational bottleneck for the basic form of the algorithm. Finding ways to avoid them, or alternatively to replace them with less expensive computations, is the key to improving the efficiency of Gröbner basis calculation. The algorithms we discuss in this chapter apply several different approaches to achieve greater efficiency. Some of them use Gröbner bases of homogeneous ideals or ideas inspired by the special properties of Gröbner bases in that case. So we begin in §1 by showing that the computation of a homogeneous Gröbner basis can be organized to proceed degree by degree. This gives the framework for Traverso’s Hilbert driven Buchberger algorithm, discussed in §2, which uses the Hilbert function of a homogeneous ideal to control the computation and bypass many unnecessary S-polynomial remainder calculations. We also show in §1 that the information generated by several S-polynomial remainder computations can be obtained simultaneously via row operations on a suitable matrix. This connection with linear algebra is the basis for Faugère’s F4 algorithm presented in §3. Finally, we introduce the main ideas behind signature-based Gröbner basis algorithms, including Faugère’s F5 algorithm, in §4.
David A. Cox, John Little, Donal O’Shea
Correction to: Ideals, Varieties, and Algorithms
David A. Cox, John Little, Donal O’Shea
Backmatter
Metadaten
Titel
Ideals, Varieties, and Algorithms
verfasst von
David A. Cox
John Little
Donal O’Shea
Copyright-Jahr
2015
Verlag
Springer International Publishing
Electronic ISBN
978-3-319-16721-3
Print ISBN
978-3-319-16720-6
DOI
https://doi.org/10.1007/978-3-319-16721-3

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