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Efficient Algorithms and Bounds for Wu-Ritt Characteristic Sets

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Effective Methods in Algebraic Geometry

Part of the book series: Progress in Mathematics ((PM,volume 94))

Abstract

The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu Wen-Tsiin, in the late seventies. Since then Wu-Ritt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated Theorem Proving in Elementary Geometries and Computer Vision. The original algorithm of Ritt, and subsequent modifications by Wu, has a non-elementary worst-case time complexity, and could be used for computing only an extended characteristic set. In this paper, we present optimal algorithms for computing a characteristic set with simple-exponential sequential and polynomial parallel time complexities. These algorithms are derived, via linear algebra, from simple-exponential degree bounds for a characteristic set. The degree bounds are obtained by using the recent effective version of Hilbert’s Nullstellensatz, due to D. Brownawell and J. Kollár, and a version of Bezout’s Inequality, due to J. Heintz.

supported by Italian Government Graduate Fellowship

Supported by NSF Grant #DMS-87-03458, ONR Grant #N00014-89-J3042

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References

  1. A. V. Aho, J. E. Hopcroft, J. D. Ullman, “The Design and Analysis of Computer Algorithms,” Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.

    MATH  Google Scholar 

  2. D. Bayer, The Division Algorithm and the Hilbert Scheme, PhD. Harvard University, 1982.

    Google Scholar 

  3. D. Bayer and M. Stillman, A Criterion for Detecting m-Regularity, Inventiones Mathematicae 87 (1987), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Bayer and M. Stillman, On the Complexity of Computing Syzygies, J. of Symbolic Computation 6 (1988), 135–147.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. J. Berkowitz, On Computing the Determinant in Small Parallel Time using a Small Number of Processors, Information Processing Letters 18 (1984), 147–150.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Brownawell, Bounds for the Degree in the Nullstellensatz, Annals of Mathematics 126 (1987), 577–591.

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Buchberger, Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory, in “Recent Trends in Multidimensional System Theory,” (Ed. Bose), Reidel, 1985.

    Google Scholar 

  8. Chou Shang-Ching, “Mechanical Geometry Theorem Proving,” D. Reidel Publishing Company, Kluwer Academic Publishers Group, Dordrecht, Boston, 1988.

    MATH  Google Scholar 

  9. Chou Shang-Ching, Proving and Discovering Theorems in Elementary Geometries Using Wu’s Method, PhD.Thesis, Department of Mathematics, University of Texas at Austin, 1985.

    Google Scholar 

  10. D. Coppersmith and S. Winograd, Matrix Multiplication via Arithmetic Progressions, in “Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing,” 1987, pp. 1–6.

    Google Scholar 

  11. L. Csanky, Fast Parallel Matrix Inversion Algorithms, SIAM Journal of Computing 5 (1976), 618–623.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Dickenstein, N. Fitchas, M. Giusti and C. Sessa, The Membership Problem for Unmixed Polynomial Ideals is Solvable in Subexponential Time, 1989, (Unpublished Manuscript).

    Google Scholar 

  13. T. Dubé, B. Mishra and Chee-keng Yap, Complexity of Buchberger’s algorithm for Grobner bases, Extended Abstract, 1986.

    Google Scholar 

  14. T. Dubé, Quantitative Analysis Problems in Computer Algebra: Grobner Bases and the Nullstellensatz, PhD. Courant Institute of Mathematical Sciences, New York University”, 1989.

    Google Scholar 

  15. G. Gallo, La Dimostrazione Automática in Geometria e Questioni di Complessita’ Correlate, Tesi di Dottorato di Ricerca, Catania, 1989.

    Google Scholar 

  16. M. Giusti, Some Effectivity Problems in Polynomial Ideal Theory, in “EUROSAM 84,” Lecture Notes in Computer Science, 174, Springer-Verlag, 1984, pp. 159–171.

    Article  MathSciNet  Google Scholar 

  17. J. Heintz, Definability and Fast Quantifier Elimination over Algebraically Closed Fields, Theoretical Computer Science 24 (1983), 239–277.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. T. Huynh, A Superexponential Lower Bound for Gröbner Bases and Church-Rosser Commutative Thue Systems, Information and Control 86 (1986), 196–206.

    Article  MathSciNet  Google Scholar 

  19. E. R. Kolchin, “Differential Algebra and Algebraic Groups,” Academic Press, New York, 1973.

    MATH  Google Scholar 

  20. J. Kollár, Sharp Effective Nullstellensatz, Unpublished Manuscript, 1988.

    Google Scholar 

  21. B. Kutzler and S. Stifter, Automated Geometry Theorem Proving Using Buchberger’s Algorithm, in “Proceedings of the 1986 Symposium on Symbolic and Algebraic Computation,” 1986, pp. 209–214.

    Google Scholar 

  22. D. Lazard, Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations, in “Computer Algebra Proceedings,” Lecture Notes in Computer Science, 162, Springer-Verlag, 1983, pp. 146–157.

    Article  MathSciNet  Google Scholar 

  23. E. W. Mayr and A. R. Meyer, The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals, Advances in Mathematics 46 (1982), 305–329.

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Mishra, “Algorithmic Algebra,” Accepted for publication by Springer-Verlag in the Texts and Monographs in Computer Science Series, 1989.

    Google Scholar 

  25. B. Mishra and Chee-keng Yap Notes on Gröbner Bases, Information Sciences 48 (1989), 219–252.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. M. Möller and F. Mora, Upper and Lower Bounds for the Degree of Gröbner Bases, in “EUROSAM 84,” Lecture Notes in Computer Science, 174, Springer-Verlag, 1984, pp. 172–183.

    Article  Google Scholar 

  27. K. Mulmuley, A Fast Parallel Algorithm to Compute the Rank of a Matrix over an Arbitrary Field, Combinatorica 7 (1987), 101–104.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. F. Ritt, “Differential Algebra,” American Mathematical Society, New York, 1950.

    MATH  Google Scholar 

  29. Wu Wen-Tsiin, On the Decision Problem and the Mechanization of Theorem Proving in Elementary Geometry, Scientia Sinica 21 (1978), 157–179.

    Google Scholar 

  30. Wu Wen-Tsiin, Basic principles of Mechanical Theorem Proving in Geometries, Journal of Sys. Sci. and Math. Sci. 4 (1984), 207–235; Also in, Journal of Automated Reasoning 2 (1986), 221-252.

    Google Scholar 

  31. Wu Wen-Tsiin, Some Recent Advances in Mechanical Theorem-Proving of Geometries, in “Automated Theorem Proving: After 25 Years,” Contemporary Mathematics, 1984, pp. 235–242 American Methematical Society.

    Google Scholar 

  32. Chee-keng Yap, A Double-Exponential Lower Bound for Degree-Compatible Gröbner Bases, NYU-Courant Robotics Laboratory Report, n.181, 1988.

    Google Scholar 

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Gallo, G., Mishra, B. (1991). Efficient Algorithms and Bounds for Wu-Ritt Characteristic Sets. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_8

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  • DOI: https://doi.org/10.1007/978-1-4612-0441-1_8

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6761-4

  • Online ISBN: 978-1-4612-0441-1

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