Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Stabilization of polynomial systems solving with Groebner bases
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Systems of Algebraic Equations Solved by Means of Endomorphisms
AAECC-10 Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A New Criterion for Normal Form Algorithms
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A complete symbolic-numeric linear method for camera pose determination
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
Generalized normal forms and polynomial system solving
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Border bases of positive dimensional polynomial ideals
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Stable border bases for ideals of points
Journal of Symbolic Computation
Computing a structured Gröbner basis approximately
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Selecting lengths of floats for the computation of approximate Gröbner bases
Journal of Symbolic Computation
Structures of precision losses in computing approximate Gröbner bases
Journal of Symbolic Computation
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Artificial discontinuity is a kind of singularity at a parametric point in computing the Grobner 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 Grobner basis will still have a jump at the parametric point. This phenomenon can cause instabilities in computing approximate Grobner 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.