Abstract
In his famous 1988 book “Mechanical Geometry Theorem Proving”, Shang-Ching Chou details a method with two steps to check whether a geometric theorem is “generally true” or not using Groebner bases. The second step consists of checking the membership of the thesis polynomial to the radical (J) of a certain ideal (L). Chou mentions: “However, for all theorems we have found in practice, J=L.” In his 2007 book “Selected topics in geometry with classical vs. computer proving”, Pavel Pech shows a beautiful example where checking the radical membership, not only the ideal membership, is required. Using a kind of “reverse engineering” we shall show how to easily find examples of theorems where to check the radical membership is required. The idea is just to construct a thesis involving an ideal such that the ideal of its variety is not itself (therefore, the ideal is not equal to its radical).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bazzotti, L., Dalzotto, G., Robbiano, L.: Remarks on geometric theorem proving. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 104–128. Springer, Heidelberg (2001)
Botana, F., Valcarce, J.L.: A software tool for the investigation of plane loci. Mat. Comp. Simul. 61(2), 141–154 (2003)
Botana, F.: A web-based resource for automatic discovery in plane geometry. Intl. J. Comp. Mat. Learning 8(1), 109–121 (2003)
Buchberger, B: Applications of Gröbner bases in non-linear computational geometry, in: Rice, J.R. (ed.) Mathematical Aspects of Scientific Software, pp. 59–87. Springer (1988)
Bulmer, M., Fearnley-Sander, D., Stokes, T.: The kinds of truth of geometry theorems. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 129–142. Springer, Heidelberg (2001)
Chou, S.-C.: Mechanical geometry theorem proving. Reidel, Dordrecht (1988)
Conti, P., Traverso, C.: Algebraic and Semialgebraic Proofs: Methods and Paradoxes. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 83–103. Springer, Heidelberg (2001)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, New York (1997)
Kutzler, B.: Careful algebraic translations of geometry theorems. In: Gonnet, G.H. (ed.) Proc. ISSAC 1989, pp. 254–263. ACM Press, Portland (1989)
Kutzler, B., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comp. 2(4), 389–397 (1986)
Manubens, M., Montes, A.: Minimal Canomical Comprehensive Groebner system. J. Symb. Comput. 44, 463–478 (2006)
Montes, A., Recio, T.: Automatic discovery of geometry theorems using minimal canonical comprehensive gröbner systems. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 113–138. Springer, Heidelberg (2007)
Pech, P.: On the need of radical ideals in automatic proving: A theorem about regular polygons. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 157–170. Springer, Heidelberg (2007)
Pech, P.: Selected topics in geometry with classical vs. computer proving. World Scientific Publishing Co. Pte. Ltd, Singapore (2007)
Recio, T., Botana, F.: Where the truth lies (in automatic theorem proving in elementary geometry). In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) ICCSA 2004. LNCS, vol. 3044, pp. 761–770. Springer, Heidelberg (2004)
Recio, T., Vélez, M.P.: An Introduction to Automated Discovery in Geometry through Symbolic Computation. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation, vol. 1, pp. 257–271. Springer, Vienna (2012)
Roanes-Lozano, E., Roanes-Macías, E., Villar-Mena, M.: A bridge between dynamic geometry and computer algebra. Mat. Comp. Mod. 37(9-10), 1005–1028 (2003)
Roanes-Lozano, E., Roanes-Macías, Laita, L.M.: The geometry of algebraic systems and their exact solving using Groebner bases. Comp. Sci. Eng. 6(2), 76–79 (2004)
Roanes-Lozano, E., Roanes-Macías, Laita, L.M.: Some Applications of Groebner bases. Comp. Sci. Eng. 6(3), 56–60 (2004)
Roanes-Macías, E., Roanes-Lozano, E.: Nuevas Tecnologías en Geometría. Editorial Complutense, Madrid (1994)
Roanes-Macías, E., Roanes-Lozano, E.: A Maple package for automatic theorem proving and discovery in 3D-geometry. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 171–188. Springer, Heidelberg (2007)
Roanes-Macías, E., Roanes-Lozano, E., Fernández-Biarge, J.: Extensión natural a 3D del teorema de Pappus y su configuración completa. Bol. Soc. “Puig Adam” 80, 38–56 (2008)
Roanes-Macías, E., Roanes-Lozano, E., Fernández-Biarge, J.: Obtaining a 3D extension of Pascal theorem for non-degenerated quadrics and its complete configuration with the aid of a computer algebra system. RACSAM (Rev. R. Acad. C. Exactas, Fís. Nat., Serie A, Mat.) 103(1), 93–109 (2009)
Roanes-Macías, E., Roanes-Lozano, E.: Un método algebraico-computacional para demostración automática en geometría euclídea. Bol. Soc. “Puig Adam” 88, 31–63 (2011)
Sun, Y., Wang, D., Zhou, J.: A new method of automatic geometric theorem proving and discovery by comprehensive Groebner systems. In: Proc. ADG 2012 (2012) (to appear), http://dream.inf.ed.ac.uk/events/adg2012/uploads/proceedings/ADG2012-proceedings.pdf#page=163
Wang, D.: GEOTHER 1.1: Handling and proving geometric theorems automatically. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 194–215. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Roanes-Lozano, E., Roanes-Macías, E. (2013). A Note on the Need for Radical Membership Checking in Mechanical Theorem Proving in Geometry. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2013. Lecture Notes in Computer Science, vol 8136. Springer, Cham. https://doi.org/10.1007/978-3-319-02297-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-02297-0_24
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02296-3
Online ISBN: 978-3-319-02297-0
eBook Packages: Computer ScienceComputer Science (R0)