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Computing parametric geometric resolutions. (English) Zbl 1058.68123
Given 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.

##### 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
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