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1992 | Buch

Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

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

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

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Über dieses Buch

We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960's, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems. It is our belief that the growing importance of these computational techniques warrants their introduction into the undergraduate (and graduate) mathematics curricu­ lum. Many undergraduates enjoy the concrete, almost nineteenth century, flavor that a computational emphasis brings to the subject. At the same time, one can do some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory and the Nullstellensatz. The mathematical prerequisites of the book are modest: the students should have had a course in linear algebra and a course where they learned how to do proofs. Examples of the latter sort of course include discrete math and abstract algebra.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Geometry, Algebra, and Algorithms
Abstract
This chapter will introduce some of the basic themes of the course. 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[x l,...,x n]. Finally, we will discuss polynomials in one variable to illustrate the role played by algorithms.
David Cox, John Little, Donal O’Shea
Chapter 2. Groebner Bases
Abstract
In Chapter 1, we have seen how the algebra of the polynomial rings k[x 1,…, x n ] and the geometry of affine algebraic varieties are linked. In this chapter, we will study the method of Groebner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. The method of Groebner 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 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 Groebner bases and the classical theory of resultants. The geometric interpretation of elimination will also be explored when we discuss the Closure Theorem. Of the many applications of elimination theory, we will treat two in detail: the implicitization problem and the envelope of a family of curves.
David 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. In keeping with the computational emphasis of this course, we will develop algorithms to carry out the algebraic operations. In §§5 and 6, we will study the more important algebraic and geometric concepts arising out of the Hilbert Basis Theorem: notably the possibility of decomposing a variety into a union of simpler varieties and the corresponding algebraic notion of writing an ideal as an intersection of simpler ideals.
David Cox, John Little, Donal O’Shea
Chapter 5. Polynomial and Rational Functions on a Variety
Abstract
One of the unifying themes of modem 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 Cox, John Little, Donal O’Shea
Chapter 6. Robotics and Automatic Geometric Theorem Proving
Abstract
In this chapter we will consider two recent 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 Cox, John Little, Donal O’Shea
Chapter 7. Invariant Theory of Finite Groups
Abstract
Invariant theory has had a profound affect on the development of algebraic geometry. For example, the Hilbert Basis Theorem and Hilbert Nullstellensatz, which play a central role in the earlier chapters of this book, were proved by Hilbert in the course of his investigations of invariant theory.
David 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 k n . In this chapter, we will enlarge k n by adding certain “points at ∞” to create n-dimensional projective space ℙ n (k). We will then define projective varieties in ℙ n (k). and study the projective version of the algebra—geometry correspondence. 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 from Chapter 3. The chapter will end with a discussion of the geometry of quadric hypersurfaces.
David 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 Cox, John Little, Donal O’Shea
Backmatter
Metadaten
Titel
Ideals, Varieties, and Algorithms
verfasst von
David Cox
John Little
Donal O’Shea
Copyright-Jahr
1992
Verlag
Springer New York
Electronic ISBN
978-1-4757-2181-2
Print ISBN
978-1-4757-2183-6
DOI
https://doi.org/10.1007/978-1-4757-2181-2