Artificial discontinuities of single-parametric Gröbner bases

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.