Automated geometry theorem proving using Buchberger's algorithm

Recently, geometry theorem proving has become an important topic of research in symbolic computation. In this paper we present a new approach to automated geometry theorem proving that is based on Buchberger's Gröbner bases method, one of the most important general purpose methods in computer algebra. The goal is to automatically prove geometry theorems whose hypotheses and conjecture can be expressed algebraically, i.e. in form of polynomial equations. After shortly reviewing the basic notions of Gröbner bases and discussing some new aspects on confirming theorems, we describe two different methods for applying Buchberger's algorithm to geometry theorem proving, each of them being more efficient than the other on a certain class of problems. The second method requires a new notion of reduction, which we call pseudoreduction. This pseudoreduction yields results on polynomials over some rational function field by computations that are done merely over the rationals and, therefore, is of general interest. Finally, a computing time statistics on about 40 non-trivial examples is given, based on an implementation of the methods in the computer algebra system SAC-2 on an IBM 4341.