An Introduction to Gröbner BasesA very carefully crafted introduction to the theory and some of the applications of Gröbner bases … contains a wealth of illustrative examples and a wide variety of useful exercises, the discussion is everywhere well-motivated, and further developments and important issues are well sign-posted … has many solid virtues and is an ideal text for beginners in the subject … certainly an excellent text. —Bulletin of the London Mathematical Society As the primary tool for doing explicit computations in polynomial rings in many variables, Gröbner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Gröbner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Gröbner bases for polynomials with coefficients in a field, applications of Gröbner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Gröbner bases in modules, and the theory of Gröbner bases for polynomials with coefficients in rings. With over 120 worked-out examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra. |
Contents
| 1 | |
Chapter 2 Applications of Gröbner Bases | 53 |
Chapter 3 Modules and Gröbner Bases | 113 |
Chapter 4 Gröbner Bases over Rings | 201 |
Appendix A Computations and Algorithms | 275 |
Appendix B Wellordering and Induction | 277 |
| 279 | |
List of Symbols | 283 |
| 285 | |
Back Cover | 290 |
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Common terms and phrases
A-module assume bases basis G Buchberger's Algorithm coefficients columns compute a Gröbner coordinates Corollary corresponding coset define definition deglex denote divides Division Algorithm element elimination order equations Exercise exists finite given Grabner basis hence Hilbert Basis Theorem Hom(M homomorphism ideal of k[x1 ideal quotient integer integral domain intersection irreducible isomorphism k-algebra leading power product leading term Lemma Let f Let G lex ordering lex term ordering linear combination lm(g loop lp(f lp(g lp(gi lp(h lt(f lt(g lt(h matrix maximal ideals minimal polynomial monomials NG(f Noetherian Noetherian ring non-zero polynomials notation obtain polynomial f polynomial rings primary decomposition prime ideal Proposition Prove reduced Gröbner basis reduced with respect respect to G S-polynomials saturated set saturated subsets Section solution strong Grabner basis submodule syzygy module variables larger vectors zero