A generalized Euclidean algorithm for geometry theorem proving

Abstract

In the first part of this paper, we present an algorithm that computes an unmixed-dimensional decomposition of a varietyV. EachVi in the decompositionV=V1U...UVm is given by a finite set of polynomials which represents the generic points of the irreducible components ofVi. The basic operation in our algorithm is the computation of greatest common divisors of univariate polynomials over extension fields given by regular chains. No factorization is needed. In the second part, this algorithm is applied to geometry theorem proving. We show that it can be used for deciding whether geometry statements are generically true or whether they are true under given nondegeneracy conditions. If a geometry statement is generically true, the simplest nondegeneracy condition with respect to a lexicographical degree ordering can be constructed by means of our algorithm.