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.
In this paper, we show that with the state-of-art module intersection IBP reduction method and our improved Leinartas' algorithm, IBP relations for very complicated Feynman integrals can be solved and the analytic reduction coefficients can be dramatically simplified. We develop a large scale parallel implementation of our improved Leinartas' algorithm, based on the Singular/GPI-Space framework. We demonstrate our method by the reduction of two-loop five-point Feynman integrals with degree-five numerators, with a simple and sparse IBP system. The analytic reduction result is then greatly simplified by our improved Leinartas' algorithm to a usable form, with a compression ratio of two order of magnitudes. We further discover that the compression ratio increases with the complexity of the Feynman integrals.
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