An Introduction to Grobner BasesAmerican Mathematical Soc., Jul 21, 1994 A very carefully crafted introduction to the theory and some of the applications of Grobner 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, Grobner 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 Grobner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Grobner bases for polynomials with coefficients in a field, applications of Grobner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Grobner bases in modules, and the theory of Grobner 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. |
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3x²y A-module assume basis G Buchberger's Algorithm coefficients compute a Gröbner coordinates Corollary coset define definition deglex denote divides lp(f Division Algorithm element elimination order equations Exercise f₁ finite g₁ h₁ hence Hilbert Basis Theorem Hom(M homogeneous homomorphism ideal quotient integral domain irreducible lc(f leading power product leading term Lemma Let f Let G lex ordering lex term ordering lm(g lp(fi lp(g lp(gi lt(f lt(g lt(h M₁ matrix maximal ideals minimal polynomial monomials NG(f Noetherian Noetherian ring non-zero polynomials notation obtain polynomial f polynomial rings primary decomposition prime ideal PROOF Proposition Prove reduced Gröbner basis reduced with respect respect to G S-polynomials saturated set saturated subsets Section solution strong Gröbner basis submodule syzygy module Theorem vectors x²y xy² y²x zero
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Page 7 - Therefore, to complete the proof of the Theorem, it suffices to show that condition (2) implies conditions (1) and (4).