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A subresultant-like theory for Buchberger’s procedure

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Abstract

By restricting Buchberger’s procedure so as to match practical ones and extending the structure of subresultant considerably, we develop a subresultant-like theory for Buchberger’s procedure of Gröbner basis computation. As an application of the theory, we clarify the mechanism of main-term cancellation which is the main origin of instability of the computation of Gröbner bases with floating-point numbers.

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References

  1. Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Multidimensional Systems Theory, Chap. 6. Reidel Publ., Dordrecht (1985)

  2. Collins J.E.: Subresultant and reduced polynomial remainder sequence. J. ACM 14, 128–142 (1967)

    Article  MATH  Google Scholar 

  3. Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Springer, New York (1997)

    Google Scholar 

  4. Faugère J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). J. Pure Appl. Algebra 139, 61–88 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hearn, A.C. : (1979) Non-modular computation of polynomial GCDs using trial division. In: Proceedings of EUROSAM 79. LNCS, vol. 72, pp. 227–239. Springer, Berlin

  6. Lazard, D.: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations. In: Proceedings of EUROCAL 1983. LNCS, vol. 162, pp. 146–156. Springer, Berlin (1983)

  7. Loos, R.: Generalized polynomial remainder sequences. In: Buchberger, B., Collins, G.E., Loos, R. (eds.) Computer Algebra, Symbolic and Algebraic Computation. Springer, Berlin (1983)

  8. Mandache, A.M.: The Gröbner basis algorithm and subresultant theory. In: Proceedings of ISSAC’94 (Intn’l Symposium on Symbolic and Algebraic Computation), pp. 123–128. ACM Press, New York (1994)

  9. Sasaki, T.: A practical method for floating-point Gröbner basis computation. In: Proceedings of Joint Conf. of ASCM 2009 and MACIS 2009. COE Lecture Note, vol. 22, pp. 167–176, Kyushu Univ. (2009)

  10. Sasaki T., Furukawa A.: Secondary polynomial remainder sequence and an extension of subresultant theory. J. Inf. Proces. 7, 175–184 (1984)

    MATH  MathSciNet  Google Scholar 

  11. Sasaki T., Furukawa A.: Theory of multiple polynomial remainder sequence. Publ. RIMS (Research Inst. for Math. Sci., Kyoto Univ. 20, 367–399 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sasaki, T., Kako, F.: Computing floating-point Gröbner base stably. In: Proceedings of SNC2007 (Symbolic Numeric Computation), pp. 180–189, London, Canada (2007)

  13. Sasaki, T., Kako, F.: Floating-point Gröbner basis computation with ill-conditionedness estimation. In: Kapur, D. (ed.) Proceedings of ASCM2007 (Asian Symposium on Computer Mathematics). LNAI, vol. 5081, pp. 278–292. Springer, Berlin (2008)

  14. Sasaki, T., Kako, F.: Term cancellations in computing floating-point Gröbner bases. In: Proceedings of CASC2010 (Computer Algebra in Scientific Computing). LNCS, vol. 6244, pp. 220–231. Springer, Berlin (2010)

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Authors and Affiliations

  1. Professor Emeritus, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

    Tateaki Sasaki

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  1. Tateaki Sasaki
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Correspondence to Tateaki Sasaki.

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This work was supported in part by Japan Society for the Promotion of Science under Grants 23500003.

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Sasaki, T. A subresultant-like theory for Buchberger’s procedure. Japan J. Indust. Appl. Math. 31, 137–164 (2014). https://doi.org/10.1007/s13160-013-0133-1

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  • DOI: https://doi.org/10.1007/s13160-013-0133-1

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