Computing Parametric Geometric Resolutions
Computing Parametric Geometric Resolutions
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DOI：
10.1007/s00200-002-0109-x
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发表时间：
2003-02
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通讯作者：
É. Schost
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文献类型：
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作者：
É. Schost
给定一个未知数的多项式方程组，它依赖于一些参数，我们定义了参数几何归结的概念，作为一种手段来表示一些通有解的参数.这个归结的系数是参数的有理函数，我们首先证明了它们的次数是有界的Bézout数dn，其中dn是输入系统的次数的一个界.然后，我们提出了一个概率算法来计算参数分辨率。它的复杂性是多项式的大小的输出和输入系统的评估的复杂性。成功的概率由Bézout数中的数量多项式控制，我们给出了这个过程的几个应用，特别是计算超椭圆曲线的Jacobian和真实的几何问题.
Given a polynomial system ofnequations innunknowns that depends on some parameters, we define the notion ofparametric geometric resolutionas a means to represent some generic solutions in terms of the parameters.The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout numberdn, wheredis a bound on the degrees of the input system. Then we present a probabilistic algorithm to compute a parametric resolution. Its complexity is polynomial in the size of the output and in the complexity of evaluation of the input system. The probability of success is controlled by a quantity polynomial in the Bézout number.We present several applications of this process, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry.