Mathematics in Computer Science

, Volume 5, Issue 2, pp 179–194 | Cite as

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

Article

Abstract

It is well known that in the computation of Gröbner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a new variable to rename the leading term each time we detect a “problem”. We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zero-dimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Gröbner basis method so that we can use existing efficient algorithms (for instance the F5 algorithm) to compute an approximate Gröbner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial \({\epsilon}\)-discontinuities. Experiments show that these algorithms are effective for some nontrivial problems.

Keywords

Approximate Gröbner basis Artificial discontinuity Monomial basis F5 algorithm 

Mathematics Subject Classification (2000)

Primary I.1.2 Secondary F.2.1 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.INRIA, Paris-Rocquencourt Center, SALSA Project, UPMC, Univ Paris 06, CNRS, UMR 7606ParisFrance
  2. 2.LMIB, School of Mathematics and Systems SciencesBeihang UniversityBeijingChina
  3. 3.KLMM, Institute of Systems Science, Academy of Mathematics and System ScienceChinese Academy of SciencesBeijingChina

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