Abstract
This paper presents methods for zero and ideal decomposition of partial differential polynomial systems and the application of these methods and their implementations to deal with problems from the local theory of surfaces. We show how to prove known geometric theorems and to derive unknown relations automatically. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z. Li. Moreover, we provide symmetric expressions for Li’s relation and clarify his statement. Some examples of theorem proving and computational difficulties encountered in our experiments are also discussed.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Acknowledgements
Part of this work has been supported by the SPACES Project (http://wwwspaces. lip6.fr/) and by the Chinese National 973 Project NKBRSF G19980306.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aubry, P. (1999). Ensembles triangulaires de polynômes et résolution de systémes algébriques. Implantation en Axiom. Ph.D. thesis, Université Paris VI, France.
Aubry, P., Lazard, D., Moreno Maza, M. (1999). On the theories of triangular sets. J. Symb. Comput. 28: 105–124.
Boulier, F., Lazard, D., Ollivier, F., Petitot, M. (1995). Representation for the radical of a finitely generated differential ideal. In: Levelt, A. H. M. (ed.): Proc. ISSAC’ 95, Montreal, Canada, ACM Press, New York, pp. 158–166.
Boulier, F., Lazard, D., Ollivier, F., Petitot, M. (1998). Computing representation for radicals of finitely generated differential ideals. Preprint, LIFL, Universit’ Lille I, France.
Boulier, F., Lemaire, F. (2000). Computing canonical representatives of regular differential ideals. In: Traverso, C. (ed.): Proc. ISSAC’ 2000, St. Andrews, Scotland, ACM Press, New York, pp. 38–47.
Buchberger, B. (1985). Gröbner bases: An algorithmic method for polynomial ideal theory. In: Bose, N. K. (ed): Multidimensional Systems Theory, D. Reidel, Dordrecht, pp. 184–232.
Carrà Ferro, G., Gallo, G. (1990). A procedure to prove statements in differential geometry. J. Automat. Reason. 6: 203–209.
Chou, S.-C., Gao, X.-S. (1993). Automated reasoning in differential geometry and mechanics using the characteristic set method-Part I. An improved version of Ritt-Wu’s decomposition algorithm. Part II. Mechanical theorem proving. J. Automat. Reason. 10: 161–189.
Hubert, E. (1999). The diffalg package. http://daisy.uwaterloo.ca/~ehubert/Diff alg/.
Hubert, E. (2000). Factorization-free decomposition algorithms in differential algebra. J. Symb. Comput. 29: 641–662.
Kalkbrener, M. (1991). Three Contributions to Elimination Theory. Ph.D. thesis, Johannes Kepler University, Austria.
Kalkbrener, M. (1998). Algorithmic properties of polynomial rings. J. Symb. Comput. 26: 525–581.
Klingenberg, W. (1978). A Course in Differential Geometry. Translated by D. Hoffman. Springer, New York.
Kreyszig, E. (1968). Introduction to Differential Geometry and Riemannian Geometry. University of Toronto Press, Tornoto.
Li, H. (1997). Mechanical theorem proving in differential geometry-Local theory of surfaces. Sci. China (Ser. A) 40: 350–356.
Li, H., Cheng, M. (1998). Clifford algebraic reduction method for automated theorem proving in differential geometry. J. Automat. Reason. 21: 1–21.
Li, Z. (1995). Mechanical theorem proving in the local theory of surfaces. Ann. Math. Artif. Intell. 13: 25–46.
Li, Z., Wang, D. (1999). Coherent, regular and simple systems in zero decompositions of partial differential systems. Syst. Sci. Math. Sci. 12 (Suppl.): 43–60.
Ritt, J. F. (1950). Differential Algebra. Amer. Math. Soc., New York.
Rosenfeld, A. (1959). Specializations in differential algebra. Trans. Amer. Math. Soc. 90: 394–407.
Seidenberg, A. (1956). An elimination theory for differential algebra. Univ. California Publ. Math. (N.S.) 3(2): 31–66.
Wang, D. (1995). A method for proving theorems in differential geometry and mechanics. J. Univ. Comput. Sci. 1: 658–673.
Wang, D. (1996). An elimination method for differential polynomial systems I. Syst. Sci. Math. Sci. 9: 216–228.
Wang, D. (1998). Decomposing polynomial systems into simple systems. J. Symb. Comput. 25: 295–314.
Wang, D. (2000). Automated reasoning about surfaces (progress report). In: Richter-Gebert, J., Wang, D. (eds.): Proc. ADG 2000, Zurich, Switzerland, September 25-27, 2000, pp. 183–196.
Wu, W.-t. (1979). On the mechanization of theorem-proving in elementary differential geometry (in Chinese). Sci. Sinica Special Issue on Math. (I): 94–102.
Wu, W.-t. (1987). A constructive theory of differential algebraic geometry based on works of J. F. Ritt with particular applications to mechanical theorem-proving of differential geometries. In: Gu, C., Berger, M., Bryant, R. L. (eds.): Differential Geometry and Differential Equations, LNM 1255, Springer, Berlin, pp. 173–189.
Wu, W.-t. (1989). On the foundation of algebraic differential geometry. Syst. Sci. Math. Sci. 2: 289–312.
Wu, W.-t. (1991). Mechanical theorem proving of differential geometries and some of its applications in mechanics. J. Automat. Reason. 7: 171–191.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aubry, P., Wang, D. (2001). Reasoning about Surfaces Using Differential Zero and Ideal Decomposition. 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_10
Download citation
DOI: https://doi.org/10.1007/3-540-45410-1_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42598-4
Online ISBN: 978-3-540-45410-6
eBook Packages: Springer Book Archive