We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over $\mathbb{C}$. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of $R/I$ and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for $R/I$. This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.

我们考虑以下问题：找到多项式函数集的孤立公共根，该多项式函数定义多项式的环R中的零维理想I$\mathbb{C}$。范式算法提供了一种代数方法来解决此问题。Telen等人提出的框架。（2018）使用截短范式（TNFs）计算$\mathrm{[R}/\mathrm{一世}$和I的解决方案。与标准的Monomials相比，该框架允许使用更多的通用基础$\mathrm{[R}/\mathrm{一世}$。本文将利用这一点介绍具有良好属性的两种特殊（非单项）基函数的用法。例如，这使我们能够将基本函数调整为I的根的预期位置。在非通用零维系统的情况下，我们还提出了有效计算TNF的算法和TNF构造的一般化算法。许多实验都揭示了TNF方法的潜力和新结果的实用性。