Abstract
Algorithms for computing Gröbner bases of polynomial ideals over integers, Gaussian integers and univariate polynomials over a field are discussed. Each of these algorithms takes an ideal specified by a finite set of polynomials as its input; the result is another finite basis of the ideal which can be used to simplify polynomials such that every polynomial in the ideal simplifies to 0 and every polynomial in the polynomial ring simplifies to a unique normal form. These algorithms are closely related to each other and they are extensions of Buchberger's algorithm for computing a Gröbner basis of polynomial ideals over a field. A general theorem exhibiting the uniqueness of a reduced Gröbner basis of an ideal, determined by the ordering used on indeterminates and other conditions, is given.
Some of the results reported in this paper will appear in Kandri-Rody's doctoral dissertation at RPI, Troy, NY.
Partially supported by NSF grant MCS-82-11621.
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8. References
Bachmair, L., and Buchberger, B., “A Simplified Proof of the Characterization Theorem for Gröbner-Bases,” ACM-SIGSAM Bulletin, 14/4, 1980, pp. 29–34.
Buchberger, B., “A Theoretical Basis for the Reduction of Polynomials to Canonical Forms,” ACM-SIGSAM Bulletin, 39, August 1976, pp. 19–29.
Buchberger, B., “A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner-Bases,” Proceedings of EUROSAM 79, Marseille, Springer Verlag Lecture Notes in Computer Science, Vol. 72, 1979, pp. 3–21.
Buchberger, B. and Loos, R., “Algebraic Simplification,” Computer Algebra: Symbolic and Algebraic Computation (B. Buchberger, G.E. Collins, and R. Loos, eds.), Computing Suppl. 4, Springer Verlag, New York, 1982, pp. 11–43.
Huet, G., “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems,” JACM, Vol. 27, No. 4, October 1980, pp. 797–821.
Huet, G., “A Complete Proof of Correctness of the Knuth-Bendix Completion Procedure,” JCSS, Vol. 23, No. 1, August 1981, pp. 11–21.
Hsiang, J., Topics in Theorem Proving and Program Synthesis, Ph.D. Thesis, University of Illinois, Urbana-Champagne, July 1983.
Kandri-Rody, A., Effective Problems in the Theory of Polynomial Ideals, Forthcoming Ph.D. Thesis, RPI, Troy, NY, May 1984.
Kandri-Rody, A. and Kapur, D., “On Relationship between Buchberger's Gröbner Basis Algorithm and the Knuth-Bendix Completion Procedure,” TIS Report No. 83CRD286, General Electric Research and Development Center, Schenectady, NY, December 1983.
Kandri-Rody, A. and Kapur, D., “Computing the Gröbner Basis of Polynomial Ideals over Integers,” to appear in Third MACSYMA User's Conference, Schenectady, NY, July 1984.
Kandri-Rody, A. and Saunders, B.D., “Primality of Ideals in Polynomial Rings,” to appear in Third MACSYMA User's Conference, Schenectady, NY, July 1984.
Kapur, D. and Narendran, P., “The Knuth-Bendix Completion Procedure and Thue Systems,” Third Conference on Foundation of Computer Science and Software Engg., Bangalore, India, December 1983, pp. 363–385.
Kapur, D. and Sivakumar, G., “Architecture of and Experiments with RRL, a Rewrite Rule Laboratory,” Proceedings of the NSF Workshop on Rewrite Rule Laboratory, Rensselaerville, NY, September 4–6, 1983.
Knuth, D.E. and Bendix, P.B., “Simple Word Problems in Universal Algebras,” Computational Problems in Abstract Algebras (J. Leech, ed.), Pergamon Press, 1970, pp. 263–297.
Lankford, D.S. and Ballantyne, A.M., Private Communication, December 1983.
Lankford, D.S. and Butler, G., “Experiments with Computer Implementations of Procedures which often Derive Decision Algorithms for the Word Problem in Abstract Algebra,” Technical Report, MTP-7, Louisiana Tech. University, August 1980.
Lauer, M., “Canonical Representatives for Residue Classes of a Polynomial Ideal,” SYMSAC, 1976, pp. 339–345.
Lausch, H., and Nobaurer, W., Algebra of Polynomials, North-Holland, Amsterdam, 1973.
Schaller, S., Algorithmic Aspects of Polynomial Residue Class Rings, Ph.D. Thesis, Computer Science Tech., University of Wisconsin, Madison, Rep. 370, 1979.
Szekeres, G., “A Canonical Basis for the Ideals of a Polynomial Domain,” American Mathematical Monthly, Vol. 59, No. 6, 1952, pp. 379–386.
van der Waerden, B.L., Modern Algebra, Vols. I and II, Fredrick Ungar Publishing Co., New York, 1966.
Zacharias, G., Generalized Gröbner Bases in Commutative Polynomial Rings, Bachelor Thesis, Lab. for Computer Science, MIT, 1978.
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Kandri-Rody, A., Kapur, D. (1984). Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings. In: Fitch, J. (eds) EUROSAM 84. EUROSAM 1984. Lecture Notes in Computer Science, vol 174. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0032842
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DOI: https://doi.org/10.1007/BFb0032842
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