Artificial discontinuities of single-parametric Gröbner bases

Jean-Charles Faugère 1 Ye Liang 1
1 SALSA - Solvers for Algebraic Systems and Applications
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
Abstract : Artificial discontinuity is a kind of singularity at a parametric point in computing the Gröbner basis of a specialized parametric ideal w.r.t. a certain term order. When it occurs, though parameters change continuously at the point and the properties of the parametric ideal have no sudden changes, the Gröbner basis will still have a jump at the parametric point. This phenomenon can cause instabilities in computing approximate Gröbner bases.In this paper, we study artificial discontinuities in single-parametric case by proposing a solid theoretical foundation for them. We provide a criterion to recognize artificial discontinuities by comparing the zero point numbers of specialized parametric ideals. Moreover, we prove that for a single-parametric polynomial ideal with some restrictions, its artificially discontinuous specializations (ADS) can be locally repaired to continuous specializations (CS) by the TSV (Term Substitution with Variables) strategy.
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Journal of Symbolic Computation, Elsevier, 2011, 46 (4), pp.459--466. 〈10.1016/j.jsc.2010.11.001〉
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Soumis le : mercredi 29 avril 2015 - 17:50:15
Dernière modification le : samedi 1 décembre 2018 - 01:27:06

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Jean-Charles Faugère, Ye Liang. Artificial discontinuities of single-parametric Gröbner bases. Journal of Symbolic Computation, Elsevier, 2011, 46 (4), pp.459--466. 〈10.1016/j.jsc.2010.11.001〉. 〈hal-01147173〉

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