Abstract
Some geometric theorems can be stated in coordinate-free form as polynomials in Grassman algebra and can be proven by the anticommutative Gröbner basis method. In this article, we analyze some properties of both sets of hypotheses and conclusions of the theorem.
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References
F. A. Beresin, The Method of Secondary Quantization [in Russian], NFMI (2000)
F. A. Beresin, Introduction to Algebra and Analysis with Anticommutative Variables [in Russian], Izd. Mosk. Univ., Moscow (1983).
B. Buchberger, “Gröbner bases: An algorithmic method in polynomial ideal theory,” in: N. K. Bose, ed., Recent Trends in Multidimensional System Theory, Reidel (1985).
S.-C. Chou, Mechanical Geometry Theorem Proving. Mathematics and Its Applications, D. Reidel, Dordrecht (1987).
S.-C. Chou, X.-S. Gao, and J.-Z. Zhang, “Automated geometry theorem proving by vector calculation,” in: M. Bronstein, ed., Proc. ISSAC ’93, ACM Press (1993), pp. 284–291.
D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Springer, New York (1998).
D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, Springer, New York (1998).
D. Eisenbud, I. Peeva, and B. Sturmfels, “Noncommutative Grobner bases for commutative ideals,” Proc. Amer. Math. Soc., 126, No. 3, 687–691 (1998).
Y. El From, Sur les Algèbres de Type Résoluble, Thèse de 3e cycle, Univ. Paris 6 (1983).
D. Hartley and P. Tuckey, “Gröbner bases in Clifford and Grassman algebras,” J. Symbolic Comput., 20, No. 2, 197–205 (1995).
A. Kandry-Rody and V. Weispfenning, “Non-commutative Gröbner bases in algebras of solvable type,” J. Symbolic Comput., 9, 1–26 (1990).
F. Mora, “Gröebner bases for non-commutative polynomial rings,” in: Proc. AAECC-5, Springer LNCS, Vol. 229 (1986), pp. 353–362.
T. Mora, “Gröbner bases in noncommutative algebras,” in: Proc. ISSAC’88, Springer LNCS, Vol. 358 (1989), pp. 150–161.
T. Mora, “An introduction to commutative and noncommutative Gröbner bases,” Theoret. Comput. Sci., 134, 131–173 (1994).
S. Stifter, “Geometry theorem proving in vector spaces by means of Gröbner bases,” in: M. Bronstein, ed., Proc. ISSAC ’93, ACM Press (1993), pp. 301–310.
I. J. Tchoupaeva, “Application of methods of noncommutative Gröbner bases to the proof of geometrical statements given in noncoordinate form,” in: Proc. International Workshop on Computer Algebra and Its Application to Physics, Dubna (2001).
I. J. Tchoupaeva, “Application of the noncommutative Gröbner bases method for proving geometric statements in coordinate free form,” in: Proc. Workshop on Under-and Overdetermined Systems of Algebraic or Differential Equation, Karlsruhe (2002).
V. Ufnarovski, “Introduction to noncommutative Gröbner bases theory,” Proc. London Math. Soc., 251, 259–280 (1998).
V. W. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer (1998).
E. B. Vinberg, A Course in Algebra [in Russian], Factorial Press, Moscow (2002).
D. Wang, “Gröbner bases applied to geometric theorem proving and discovering,” in: B. Buchberger and F. Winkler, eds., Gröbner Bases and Applications, Cambridge Univ. Press (1998), pp. 281–302.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 213–228, 2003.
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Tchoupaeva, I.J. Automated proving and analysis of geometric theorems in coordinate-free form by using the anticommutative Gröbner basis method. J Math Sci 135, 3409–3419 (2006). https://doi.org/10.1007/s10958-006-0170-2
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DOI: https://doi.org/10.1007/s10958-006-0170-2