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.

G is a Gr6bner basis for l, there exists i such that Ip(gi) divides Ip(j,). After renumbering if necessary, we may assume that i = 1. Now 91 is also in 1, and hence, since F is a Gr6bner basis for l, 48 CHAPTER 1. BASIC THEORY OF GROBNER BASES there exists j sueh that lp(fj) divides Ip(YI)' Therefore lp(fj) divides Ip(J,), and henee j = 1, sinee F is a minimal Grübner basis. Thus Ip(J,) = lp(y,). Now 12 is in J, and henee there exists i sueh that lp(y,) divides Ip(12), sinee G is a Grübner

is in fact {h, Js}, since the reduced Grübner basis is unique. EXAMPLE 1.8.9. We go back to Example 1.7.11. There we showed that a Grübner basis for I = (x 2 + y2 + 1, x 2y + 2xy + x) ç: Zs [x, y] with respect to the lex ordering with x> y is {x 2+y2+1, x 2y+2xy+x, 3xy+4x+y3+ y , 4ys+3y4+ y2+y+3}. By Corollary 1.8.3, {x 2+y2+1,xy+3x+2y3+2y, yS+2y4+4y2+4y+2} is a minimal Grübner basis for I. In fact it is easy to see that it is the reduced Grübner basis for I. ExercÎses 1.8.1. Compute the reduced

term order in 11, or it is not. In the first case, use Exercise 1.6.17 and in the second add a polynomial to JI, ... ,f, and repeat the argument.] c. Conclude that tbere are only finitely many reduced Grübner bases for a given ideal. d. A set F which is a Grübner basis for an ideal l with respect to every term order is called a universal Grobner basis. Use c to show that every ideal has a universal Grobner basis. An example of such a basis is given in Exercise 1.8.7. 1.8.7. Find a universal

course, each of these two orders could just as weil have been defined with a different ordering on the subscripts {l, ... , m}. In order to indicate which order we are using we will write, for example, el < e2 < ... < e m . There are many other examples of orders and we will use an order different frOID either one of the above in the next section. We note that we are using the symbol "<" in two different ways, bath for a term order on the power products of A and for a term order On the monomials

module case; that is, give a definition of Gr6bner bases for modules in terms of the syzygy module of the leading terms. 3.7.7. State and prove the analog of Corollary 3.3.3, and use it to describe how crit2 can be implemented in Aigorithm 3.5.2. Note that we have already seen in Exercise 3.5.16 that critl cannot be used in the module case. 3.7.8. Use Exercise 3.7.7 to state and prove the analog of Exercise 3.4.4 in the module case (generalizing Theorem 3.7.3). 3.7. SYZYGIES FOR MODULES 169