Computer Algebra and Algebraic. Geometry-Achievements and Perspectives

De computer is niet de steen maar de slijpsteen der wijzen. (The computer is not the philosopher's stone but the philosopher's whetstone.) Hugo Battus, Rekenen op taal (1989) Contents 1 Preface 2 Introduction by pictures 3 Some problems in algebraic geometry 4 Some global algorithms 5 Singularities and standard bases 6 Some local algorithms 7 Computer algebra solutions to singularity problems 8 What else is needed References In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this article is to show, by means of examples, the usefulness of computer algebra to mathematical research. Computer algebra itself is a highly diversified discipline with applications to various areas of mathematics; we find many of these in numerous research papers, in proceedings or in textbooks (cf. Here we concentrate mainly on Gröbner bases and leave aside many other topics of computer algebra (cf. Davenport, In particular, we do not mention (multivariate) polynomial factorisation, another major and important tool in computational algebraic geometry. Gröbner bases were introduced originally by Buchberger as a computational tool for testing solvability of a system of polynomial equations, to count the number of solutions (with multiplicities) if this number is finite and, more algebraically, to compute in the quotient ring modulo the given polynomials. Since then, Gröbner bases have become the major computational tool, not only in algebraic geometry. The importance of Gröbner bases for mathematical research in algebraic geometry is obvious and their use needs, nowadays, hardly any justification. Indeed, chapters on Gröbner bases and Buchberger's algorithm (Buchberger, 1965) have been incorporated in many new textbooks on algebraic geometry such as the books of Cox, Little and Computational methods become increasingly important in pure mathematics and the above mentioned books have the effect that Gröbner bases and their applications become a standard part of university courses on algebraic geometry and commutative algebra. One of the reasons is that these methods, together with very efficient computers, allow the treatment of non–trivial examples and, moreover, are applicable to non–mathematical, industrial, technical or economical problems. Another reason is that there is a belief that algorithms can contribute to a deeper understanding of a problem. The human idea of " understanding " is clearly part of the historical, cultural and technical status …