Bruno Buchberger

A criterion for detecting unnecessary reductions in the construction of Groebner bases

We present a new criterion that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals. The application of the criterion may drastically reduce the number of reductions of polynomials in the course of the algorithm. Incidentally, the new criterion allows to derive a realistic upper bound for the degrees of the polynomials in the Grobner-bases computed by the algorithm in the case of polynomials in two variables.

Computer algebra symbolic and algebraic computation

Computer algebra is an alternative and complement to numerical mathematics. Its importance is steadily increasing. This volume is the first systematic and complete treatment of computer algebra. It presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature. The volume follows a top-down structure proceeding from very high-level problems which will be well-motivated for most readers to problems whose solution is needed for solving the problems at the higher level. The volume is written as a supplementary text for a traditional algebra course or for a general algorithms course. It also provides the basis for an independent computer algebra course.

Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal

This is the English translation (by Michael P. Abramson) of the PhD thesis of Bruno Buchberger, in which he introduced the algorithmic theory of Grobner bases. Some comments by Buchberger on the translation and the thesis are given in an additional short paper in this issue of the Journal of Symbolic Computation.

A theoretical basis for the reduction of polynomials to canonical forms

We define a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of polynomial ideals (e.g. the problem of constructing canonical forms for polynomials) is reducible to the construction of bases of this type. We prove a characterization theorem for these bases which immediately leads to an effective method for their construction.

Applications of Gro¨bner bases in non-linear computational geometry

Grobner bases are certain finite sets of multivariate polynomials. Many problems in polynomial ideal theory (algebraic geometry, non-linear computational geometry) can be solved by easy algorithms after transforming the polynomial sets involved in the specification of the problems into Grobner basis form. In this paper we give some examples of applying the Grobner bases method to problems in non-linear computational geometry (inverse kinematics in robot programming, collision detection for superellipsoids, implicitization of parametric representations of curves and surfaces, inversion problem for parametric representations, automated geometrical theorem proving, primary decomposition of implicitly defined geometrical objects). The paper starts with a brief summary of the Grobner bases method.

Theorema: Towards computer-aided mathematical theory exploration

Theorema is a project that aims at supporting the entire process of mathematical theory exploration within one coherent logic and software system. This survey paper illustrates the style of Theorema-supported mathematical theory exploration by a

Some properties of Gröbner-bases for polynomial ideals

We give a uniqueness theorem for Gröbner-bases of polynomial ideals and show that it is effectively decidable whether a given basis is a (minimal normed) Gröbner-basis. Incidentally, we show how our methods may be applied to decide α ≤ β for given polynomial ideals α and β.

Gröbner Bases and Systems Theory

We present the basic concepts and results of Gröbner bases theory for readers working or interested in systems theory. The concepts and methods of Gröbner bases theory are presented by examples. No prerequisites, except some notions of elementary mathematics, are necessary for reading this paper. The two main properties of Gröbner bases, the elimination property and the linear independence property, are explained. Most of the many applications of Gröbner bases theory, in particular applications in systems theory, hinge on these two properties. Also, an algorithm based on Gröbner bases for computing complete systems of solutions (“syzygies”) for linear diophantine equations with multivariate polynomial coefficients is described. Many fundamental problems of systems theory can be reduced to the problem of syzygies computation.

A survey of the Theorema project

The Theorems project aims at extending current computer algebra systems by facilities jor supporting mathematical proving. The present early-prototype version of the Theorems software system is implemented in Mathetnatica 3.0. The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing human-readable proofs in natuml language presented in nested cells that facilitate studying the computer-generated proofs at various levels of detail. The special provers are intimately connected with the junctors that build up the various mathematical domains. 1 The Objectives of the Theorems Project The Tlaeorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a “knowledge base” K of formulae (and a logical / computational derivation mechanism L),

Introduction to Groebner Bases

A comprehensive treatment of Groebner bases theory is far beyond what can be done in four lectures. Recent text books on Groebner bases like (Becker, Weispfenning 1993) and (Cox, Little, O’Shea 1992) present the material on several hundred pages. However, there are only a few key ideas behind Groebner bases theory. It is the objective of these four lectures to explain these ideas as simply as possible and to give an overview of the applications.