Artificial discontinuities of single-parametric Gröbner bases (Q629351)

The present paper gives formal definitions of continuous, artificially and intrinsically discontinuous specializations of a system of polynomial equations whose coefficients depend on some parameters. Artificial discontinuity is a key concept in understanding and resolving unstability problems appearing in the computation of an approximate Gröbner basis for the system. One of the main results of the paper is a criterium to recognize a continuous (an artificially discontinuous) specialization in the single parametric case. The other main result says that one can locally transform an artificially discontinuous specialization to a continuous one under a mild condition; there may be new artifically discontinuous specializations possibly not disturbing the numerical computations.