Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka’s celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores.

引入并行性并探索其使用仍然是计算机代数社区的一项基本挑战。另一方面，在高性能数值仿真中，遵循分离协调和计算原理的透明分布式计算环境已经成为成功的故事。在本文中，我们探索了在计算机代数环境中使用此原理的潜力。更准确地说，我们结合了两个完善的系统：我们感兴趣的数学在计算机代数系统Singular中实现，其工作重点是多项式计算，而协调工作则由工作流管理系统GPI-Space来完成，该系统依赖于Petri网作为其数学建模语言，并且已成功地用于协调学术代码的并行执行（自动并行化）适用于地震数据处理等应用领域中的商业软件。我们努力的结果是朝着在Singular应用领域进行大规模并行计算的框架迈出了重要的一步，特别是可交换代数和代数几何。作为该框架的第一个测试案例，我们为代数形式建模并实施了混合平滑度测试，该测试将Hironaka著名的去奇化证明与经典Jacobian准则相结合。将我们的实现应用于来自代数几何当前研究的两个示例，其中一个不能用其他方式处理，我们说明了框架内平滑度测试的行为，并研究了计算如何扩展到256个核。