Fundamental Algorithms in Singular

DFG优先方案SPP1489的总体目标是推动和结合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感兴趣的中心问题。为实现这一目标，该方案旨在创建一个免费和开放源码的平台，用于连接该方案所涉领域的专门计算机代数系统。然而，首先也是最重要的是，鉴于参与研究人员的需要，必须扩展各个系统。我们的建议旨在为单一系统做到这一点，我们认为，该系统将在这方面发挥关键作用。我们的目标是提高最基本的多项式计算算法的性能，但也要重新设计一些算法，以应用于算术几何和非交换代数等领域。同时，我们还考虑了一些先进的计算工具。其中包括代数几何和凸几何中的计算方法的结合，以及上同调的计算，以及上同调的计算，这些方法在环面几何和热带几何中有潜在的应用，在Deligne-Lusztig簇和单列中有潜在的应用。另一个目标是将符号算法和数值算法联系起来，特别强调初级分解的新算法。

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.