The Kinds of Truth of Geometry Theorems

In Jürgen Richter-Gebert & Dongming Wang (eds.), LNCS: Lecture Notes In Computer Science. Springer Verlag. pp. 129-142 (2001)
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Abstract
Proof by refutation of a geometry theorem that is not universally true produces a Gröbner basis whose elements, called side polynomials, may be used to give inequations that can be added to the hypotheses to give a valid theorem. We show that (in a certain sense) all possible subsidiary conditions are implied by those obtained from the basis; that what we call the kind of truth of the theorem may be derived from the basis; and that the side polynomials may be classified in a useful way. We analyse the relationship between side polynomials and kinds of truth, and we give a unified algorithmic treatment of side polynomials, with examples generated by an implementation.
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