Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift the results to the rationals. In arxiv.org/abs/2401.11606, we present a novel, massively parallel framework for modular computations with polynomial data, which is able to cover a broad spectrum of applications in commutative algebra and algebraic geometry. We demonstrate the framework's effectiveness in Groebner basis computations over the rationals and algorithmic methods from birational geometry. In particular, we develop algorithms to compute images and domains of rational maps, as well as determining invertibility and computing inverses.
Our implementation modular-gspc is based on the Singular/GPI-Space framework, which uses the computer algebra system Singular as computational backend, while coordination and communication of parallel computations is handled by the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language. Convenient installation is realized through the package manager Spack. Relying on Petri nets, our approach provides automated parallelization and balancing of the load between computation, lifting, stabilization testing, and potential verification. We use error tolerant rational reconstruction to ensure termination as long as for a fixed computation there exist only finitely many bad primes. Via stabilization testing, our approach automatically finds with high probablity a minimal set of primes required for the successful reconstruction.
We present timings to illustrate the potential for a game changing improvement of performance over previous modular and non-modular methods. In particular, we illustrate that the approach scales very well with the number of processor cores used for the computation.

This paper is joint work with Dirk Basson, Magdaleen Marais, Mirko Rahn, and Patrick Hobihasina Rakotoarisoa.

In the ISSAC 2021 Tutorial Massively Parallel Computations in Algebraic Geometry, we give an introduction to the Singular/GPI-Space framework for massively parallel computations in computer algebra. Being based on the idea of separation of computation and coordination, this framework relies on the Petri net based workflow management system GPI-Space for the coordination layer, while the computer algebra system Singular is used as the frontend and computational backend. We illustrate the use of this approach in examples taken from the context of algebraic geometry.

This paper is joint work with Anne Frühbis-Krüger.

Multivariate partial fractioning is a powerful tool for simplifying rational function coefficients in scattering amplitude computations. Since current research problems lead to large sets of complicated rational functions, performance of the partial fractioning as well as size of the obtained expressions are a prime concern. In arxiv.org/abs/2104.06866 (Computer Physics Communications Vol. 294, 2024, 108942), we develop a large scale parallel framework pfd-parallel for multivariate partial fractioning, which implements and combines an improved version of Leinartas' algorithm and the MultivariateApart algorithm. Our approach relies only on open source software. It combines parallelism over the different rational function coefficients with parallelism for individual expressions. The implementation is based on the Singular/GPI-Space framework for massively parallel computer algebra, which formulates parallel algorithms in terms of Petri nets. The modular nature of this approach allows for easy incorporation of future algorithmic developments into our package. We demonstrate the performance of our framework by simplifying expressions arising from current multiloop scattering amplitude problems. We discover that the compression ratio increases with the complexity of the Feynman integrals. See here for the analytic result.

This paper is joint work with Dominik Bendle, Murray Heymann, Mirko Rahn, Lukas Ristau, Marcel Wittmann, Zihao Wu, Yingxuan Xu and Yang Zhang.

In arxiv.org/abs/2010.06895 (to appear in PoS (Proceedings of Science), Proceedings of "MathemAmplitude 2019" in Padova, Italy) we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, workflow management system GPI-Space in combination with the computer algebra system Singular to solve the trimmed IBP system via interpolation and efficient parallelization. We show, in particular, how to use the new plugin feature of GPI-Space to manage a global state of the computation and to efficiently handle mutable data. Moreover, a Mathematica interface to generate IBPs with restricted propagator degree, which is based on module intersection, is presented in this review.

Janko Böhm, Dominik Bendle, Wolfram Decker, Alessandro Georgoudis, Franz-Josef Pfreundt, Mirko Rahn, and Yang Zhang.

In computational algebraic geometry, fan traversals frequently occur at the interplay of algebraic and combinatorial constructions.  In this article (in progress), we design and implement a massively parallel approach to fan traversals. For torus actions on affine varieties, we develop a massively parallel algorithm for computing the associated GIT-fan. Our fan traversal forms one key substep of this algorithm, which in addition exploits symmetries of the problem. Our implementation is based on the workflow manament system GPI-Space, which uses Petri nets to model the algorithm in the coordintation layer, while in the computation layer we rely on the computer algebra system Singular. For the example of the Mori chamber decomposition of the Deligne-Mumford compactification of the moduli space of 6-pointed stable curves of genus zero, we provide timings which show an impressive parallel efficiency and scaling of the performance up to 640 cores (the maximum we have tested).

Janko Böhm, Anne Frühbis-Krüger, Christian Reinbold, and Mirko Rahn.

In the paper arxiv.org/abs/2003.13752 we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr0(3,8), and show that it refines the 15-dimensional skeleton of the Dressian Dr(3,8) with the exception of 23 special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr+(3,8) and compare them to the cluster complex of the classical Grassmannian Gr(3,8).

We also provide an extensive collection of data on the project.

Dominik Bendle, Janko Böhm, Yue Ren, and Benjamin Schröter.

In the paper arxiv.org/abs/1908.04301 (Journal of High Energy Physics 02 (2020) 79, 34 pp) we introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

Dominik Bendle, Janko Böhm, Wolfram Decker, Alessandro Georgoudis, Franz-Josef Pfreundt, Mirko Rahn, Pascal Wasser, and Yang Zhang.

The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In the note arxiv.org/abs/1811.06092 (Computeralgebrarundbrief 64 (2019) 8-13), we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra.

Janko Böhm, Anne Frühbis-Krüger, and Mirko Rahn.

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high performance numerical simulation, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In the paper arxiv.org/abs/1808.09727 (Foundations of Computational Mathematics (2020)) we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematical core algorithms are implemented in the computer algebra system Singular, whose focus is on polynomial computations, while coordination is handled by 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.

Janko Böhm, Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt, Mirko Rahn, and Lukas Ristau.

In the article arxiv.org/abs/1712.08131 (to appear in Mathematics of Computation), we present an algorithmic embedded desingularization of arithmetic surfaces bearing in mind implementability. Our algorithm is based on work by Cossart-Jannsen-Saito, though our variant uses a refinement of the order instead of the Hilbert-Samuel function as a measure for the complexity of the singularity. We particularly focus on aspects arising when working in mixed characteristics. Furthermore, we exploit the algorithm's natural parallel structure rephrasing it in terms of Petri nets for use in the parallelization environment GPI-Space with Singular as computational back-end.

Anne Frühbis-Krüger, Lukas Ristau, and Bernd Schober.