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.
Acknowledgements
We thank Professor Giuseppa Carrà Ferro for advice on the paper. The comments of anonymous referees were also very helpful. The work reported here was partially supported by Australian Research Council Large Grants A49132001 and A49331346.
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Bulmer, M., Fearnley-Sander, D., Stokes, T. (2001). The Kinds of Truth of Geometry Theorems. In: Richter-Gebert, J., Wang, D. (eds) Automated Deduction in Geometry. ADG 2000. Lecture Notes in Computer Science(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45410-1_8
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DOI: https://doi.org/10.1007/3-540-45410-1_8
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