Fundamental Algorithms in Singular

DFG优先计划SPP 1489的总体目标是推动和结合来自不同数学领域的计算机代数方法，并将产生的算法应用于理论和实践兴趣的中心问题。为了实现这一目标，该计划旨在创建一个自由和开放源码的平台，用于连接专门从事该计划所涵盖领域的计算机代数系统。然而，首先也是最重要的是，鉴于参与研究人员的需要，必须扩展个别系统。我们的建议旨在为单数系统做到这一点，我们相信，它将在这方面发挥关键作用。我们的目标是提高多项式计算的最基本算法的性能，但也要重新设计一些算法在算术几何和非交换代数等领域的应用。与此同时，我们有一些先进的计算工具。其中包括代数几何和凸几何计算方法的结合，在环形几何和热带几何中具有潜在的应用，以及上同调的计算，在delign - lusztig变异和单一性方面具有潜在的应用。另一个目标是将符号和数值算法联系起来，特别强调初等分解的新算法。

The overall goal of the DFG priority programme SPP 1489 is to push forward and combine computer algebra methods from different areas of mathematics, and to apply the resulting algorithms to central problems of theoretical and practical interest. To achieve this goal, the programme aims at creating a free and open source platform for linking computer algebra systems specializing in the areas covered by the programme. First and foremost, however, the individual systems have to be extended in view of the needs of the participating researchers. Our proposal aims at doing this for the system Singular which, as we believe, will be a key player in this context. Our goal is to improve the performance of the most fundamental algorithms for polynomial computations, but also to redesign some of the algorithms for applications in areas such as arithmetic geometry and non-commutative algebra. At the same time, we have a number of advanced computational tools in mind. These include the combination of computational methods in algebraic and convex geometry, with potential applications in toric and tropical geometry, and the computation of cohomology, with potential applications to Deligne–Lusztig varieties and monodromy. Another goal is to link symbolic and numerical algorithms, with particular emphasis on new algorithms for primary decomposition.