论文标题
对主要Artinian戒指的结果
Resultants over principal Artinian rings
论文作者
论文摘要
两个单变量多项式的结果是在交换代数中非常重要的不变性，并且在计算机代数系统中广泛使用。在这里，我们提出了一种算法，以通过Artinian Principal Rings计算它的欧几里得算法。使用相同的策略，我们展示了如何计算减少的结果和一对Bézout系数。特别注意$ \ mathbf {z}/n \ mathbf {z} $的特殊情况，我们对算法的渐近成本进行了详细的分析。最后，我们说明了如何利用算法在数字字段和多项式算术上改善$ p $ - 亚种领域的理想算术。
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of Bézout coefficient can be computed. Particular attention is devoted to the special case of $\mathbf{Z}/n\mathbf{Z}$, where we perform a detailed analysis of the asymptotic cost of the algorithm. Finally, we illustrate how the algorithms can be exploited to improve ideal arithmetic in number fields and polynomial arithmetic over $p$-adic fields.
