First Applications of Gröbner Bases

Abstract

In this chapter, we discuss some of the most immediate and important applications of Gröbner bases. The central part is formed by Sections 6.2 and 6.3, which deal with Gröbner bases in ideal theory. The theory of polynomial ideals plays an important role in algebraic geometry. There, one considers polynomials with coefficients in some field K and investigates the behavior of zeroes of these polynomials in an extension field K′ of K. (Recall that a zero of f(X1,…, X n ) is an n-tuple (a1,…, a n ) of elements of K′ with f(a1,…, a n ) = 0; cf. also Lemma 2.17 (i)). This leads to a large number of questions of an algorithmic nature, such as these: given finite bases of two ideals, what is a basis of the intersection of the latter, or, given a polynomial f and an ideal f, is it true or not that some power of f lies in I ? It has been known for a long time that all these problems can be algorithmically solved. Before the arrival of Gröbner bases, however, the complexity of these algorithms was out of bounds for all practical purposes. In this chapter, we will demonstrate how Gröbner bases provide rather straightforward solutions to many decision and construction problems in the theory of polynomial ideals. Bringing these computations within the realm of feasibility has of course stimulated vigorous mathematical research on how to further improve them. We do not attempt to capture the state of the art in the field; our aim is to lay firm mathematical foundations and to present algorithms that anyone could implement in today’s computer algebra systems. We will also use Gröbner bases in the development of the theory, thereby demonstrating that the theory of Gröbner bases is not only a powerful algorithmic method but also a cornerstone of commutative algebra.KeywordsTerm OrderWord ProblemIdeal TheoryFinite SubsetPolynomial IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.