Computing parametric geometric resolutions.

*(English)*Zbl 1058.68123Given a zero-dimensional polynomial system of \(n\) equations in \(n\) indeterminates whose coefficients depend on some parameters, the author presents an algorithm that describes its solutions in a particular way: a parametric geometric resolution (i.e., a geometric resolution whose coefficients are rational functions of the parameters involved). In a first step, some degree estimates for the parametric resolution are stated. Then, a probabilistic algorithm is constructed: by means of a formal version of Newton operator, starting from a geometric resolution of the system for a particular instance of the parameters, and taking into account the bounds for the degrees of the polynomials involved, a ‘global’ parametric geometric resolution is obtained. The probability of success of the algorithm is computed and some applications are shown. In fact, this paper deals, in a more general way, with the same problem considered in other previous papers [see for example J. Heintz, T. Krick, S. Puddu, J. Sabia and A. Waissbein, J. Complexity 16, No. 1, 70–109 (2000; Zbl 1041.65044)]. The main difference is that the hypothesis on the Noether position of the variables is dropped, allowing the appearance of rational functions, which are computed by using Padé approximants.

Reviewer: Juan Sabia (Buenos Aires)

##### MSC:

68W30 | Symbolic computation and algebraic computation |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

65H10 | Numerical computation of solutions to systems of equations |

14G15 | Finite ground fields in algebraic geometry |

14Q05 | Computational aspects of algebraic curves |