Fundamental Algorithms in Singular

DFG优先计划SPP 1489的总体目标是推动和联合收割机计算机代数方法从不同的数学领域，并将产生的算法应用于理论和实际利益的中心问题。为了实现这一目标，该方案旨在创建一个自由和开放源码平台，用于连接该方案所涵盖领域的计算机代数系统。然而，首先，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Singular系统做到这一点，正如我们所相信的那样，它将成为这方面的关键角色。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的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。这些包括在代数和凸几何的计算方法相结合，在环面和热带几何的潜在应用，和上同调的计算，与潜在的应用Deligne-Lusztig品种和monodromy。另一个目标是链接符号和数值算法，特别强调新的算法主要分解。

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.