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. 

We develop a large scale parallel implementation of our improved Leinartas' algorithm, based on the Singular/GPI-Space framework in form of the application pfd-parallel. We provide a one-line installer based on the supercomputing package manager Spack. Relying on Petri nets, we use an intertwined form of parallelism mixing parallelism over the entries of the IBP matrix and internal parallelism within the entries. 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 greatly simplified 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. See here for the analytic result.

The approach is described in arxiv.org/abs/2104.06866.

Janko Böhm, Dominik Bendle, Murray Heymann, Mirko Rahn, Lukas Ristau, Marcel Wittmann, Zihao Wu, Yingxuan Xu and Yang Zhang

In computational algebraic geometry, fan traversals frequently occur at the interplay of algebraic and combinatorial constructions.  We design and implement a massively parallel approach to fan traversals. For torus actions on affine varieties, we implement 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. The current version of the code under development is available here.

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

We develop a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries.

The approach is described in arxiv.org/abs/2003.13752.

We also provide an extensive collection of data on the project related to the tropical Grassmannian TGr(3,8).

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

We develop 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.

The approach is described in arxiv.org/abs/1908.04301 (Journal of High Energy Physics 02 (2020) 79, 34 pp) and arxiv.org/abs/2010.06895 (to appear in PoS (Proceedings of Science), Proceedings of "MathemAmplitude 2019" in Padova, Italy).

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

We implement a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. The implementation is included in the Singular/GPI-Space framework.

The approach has been described in arxiv.org/abs/1808.09727 (to appear in Foundations of Computational Mathematics).

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