Abstract
An algorithm is introduced for transforming a standard basis of a zero-dimensional ideal, in the formal power series ring, into another standard basis with respect to any given local ordering. The key ingredient of the proposed algorithm is an efficient method for solving membership problems for Jacobi ideals in local rings, that utilizes the Grothendieck local duality theorem. Namely, a new algorithm for computing a standard basis of a given zero-dimensional ideal with respect to any given local ordering, is derived by using algebraic local cohomology. Its implementation is introduced, too.
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References
Faugère, J., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. Journal of Symbolic Computation 16, 329–344 (1993)
Nabeshima, K., Tajima, S.: On efficient algorithms for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases. In: Proc. ISSAC 2014, pp. 104–111. ACM Press (2014)
Nakamura, Y., Tajima, S.: On weighted-degrees for algebraic local cohomologies associated with semiquasihomogeneous singularities. Advanced Studies in Pure Mathematics 46, 105–117 (2007)
Noro, M., Takeshima, T.: Risa/Asir- A computer algebra system. In: Proc. ISSAC 1992, pp. 387–396. ACM Press (1992)
Tajima, S., Nakamura, Y.: Algebraic local cohomology class attached to quasi-homogeneous isolated hypersurface singularities. Publications of the Research Institute for Mathematical Sciences 41, 1–10 (2005)
Tajima, S., Nakamura, Y.: Annihilating ideals for an algebraic local cohomology class. Journal of Symbolic Computation 44, 435–448 (2009)
Tajima, S., Nakamura, Y.: Algebraic local cohomology classes attached to unimodal singularities. Publications of the Research Institute for Mathematical Sciences 48, 21–43 (2012)
Tajima, S., Nakamura, Y., Nabeshima, K.: Standard bases and algebraic local cohomology for zero dimensional ideals. Advanced Studies in Pure Mathematics 56, 341–361 (2009)
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Nabeshima, K., Tajima, S. (2014). An Algorithm for Computing Standard Bases by Change of Ordering via Algebraic Local Cohomology. In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_63
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DOI: https://doi.org/10.1007/978-3-662-44199-2_63
Publisher Name: Springer, Berlin, Heidelberg
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