Ideals, Varieties, and Algorithms

1. Front Matter

   Pages i-xiii

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2. Geometry, Algebra, and Algorithms

   • David Cox, John Little, Donal O’Shea

   Pages 1-46

3. Groebner Bases

   • David Cox, John Little, Donal O’Shea

   Pages 47-111

4. Elimination Theory

   • David Cox, John Little, Donal O’Shea

   Pages 112-166

5. The Algebra-Geometry Dictionary

   • David Cox, John Little, Donal O’Shea

   Pages 167-211

6. Polynomial and Rational Functions on a Variety

   • David Cox, John Little, Donal O’Shea

   Pages 212-260

7. Robotics and Automatic Geometric Theorem Proving

   • David Cox, John Little, Donal O’Shea

   Pages 261-310

8. Invariant Theory of Finite Groups

   • David Cox, John Little, Donal O’Shea

   Pages 311-348

9. Projective Algebraic Geometry

   • David Cox, John Little, Donal O’Shea

   Pages 349-428

10. The Dimension of a Variety

    • David Cox, John Little, Donal O’Shea

    Pages 429-495

11. Back Matter

    Pages 497-538

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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.

• Dimension
• Grad
• algebraic geometry
• proof

"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

• Department of Mathematics and Computer Science, Amherst College, Amherst, USA

  David Cox

• Department of Mathematics, College of the Holy Cross, Worcester, USA

  John Little

• Department of Mathematics, Statistics, and Computer Science, Mount Holyoke College, South Hadley, USA

  Donal O’Shea

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