计算和使用极小多项式
Computing and Using Minimal Polynomials
摘要：
给定一个多项式环中的零维理想I，许多计算都是从在I中寻找一元多项式开始的。搜索I中的一元多项式是考虑P/I中元素的极小多项式的特殊情况。众所周知，极小多项式可以通过消去法计算，因此这被认为是一个“已解决的问题”。但作为许多计算的关键，它值得研究它的意义，它的优化，它的应用（例如测试零维理想是否是根，主要或最大）。我们提出了计算P/I中元素的极小多项式的有效算法。对于系数在Q中的特定情况，我们展示了如何使用模块化方法来获得有保证的结果。我们还提出了一些应用程序的极小多项式，即算法计算根式和准素分解的零维理想，也为测试的激进性和极大性。
英文摘要：
Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality.
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