ICMS 2024 - Session 5

Advancing computer algebra with massively parallel methods
22-25 July 2024, Durham, UK

Organizers

• Janko Boehm (RPTU, Germany)
• Anne Fruehbis-Krueger (Carl von Ossietzky University of Oldenburg, Germany)

Aim and Scope

Schedule

Talks

Overview Talk: Parallel Methods in Commutative Algebra and Algebraic Geometry: Past Successes and Future Challenges

Janko Boehm (RPTU, Germany), Anne Fruehbis-Krueger (University of Oldenburg, Germany)

Abstract: The introduction of massively parallel methods in commutative algebra and algebraic geometry is offering a game changing improvement of efficiency and scalability in experimental techniques as well as applied computations. This talk aims to provide an overview of the potential of parallel methods within related fields. We will look at notable examples where parallel methods have successfully been applied, and give ideas of current developments in the areas of Gröbner bases, syzygies, modular methods, birational geometry, and high energy physics. The presentation will also address ongoing efforts to establish a generic programming model, as well as potential future applications.

Using algebraic geometry to reduce Feynman integrals: automatic software NeatIBP

Yang Zhang (University of Science and Technology of China, Hefei, Anhui, China)

Abstract: We present the package NeatIBP, which automatically generates small-size integration-by-parts (IBP) identities for Feynman integrals. Based on the syzygy and module intersection techniques, the generated IBP identities' propagator degree is controlled and thus the size of the system of IBP identities is shorter than that generated by the standard Laporta algorithm. The computation powered by Singular and Mathematica. The parallelization is implemented on the level of sub-diagrams of Feynman integral via a manager-worker model. The future development will focus on the fully open-source implement and more layers of syzygy computations and Feynman integral relation searching.

Massively Parallel Modular Algorithms for the Image of Rational Maps

Hobihasina Patrick Rakotoarisoa (Stellenbosch University)

Abstract: Computations over the rational numbers frequently encounter the issue of intermediate coefficient growth. Modular methods provide a solution to this problem by applying the algorithm under consideration modulo a number of primes and then lifting the modular results to the rationals. We focus on the application of these methods within commutative algebra and algebraic geometry, showcasing their implementation in a massively parallel framework  in Groebner basis computations and various algorithmic strategies in birational geometry. Specifically, we will address  algorithms for computing the images and domains of rational maps, assessing invertibility, and deriving inverses. The effectiveness of our modular approache is evidenced through timings, which demonstrate significant performance enhancements over traditional modular and non-modular methods.

Generic modular methods for commutatve algebra and algebraic geometry

Magdaleen Marais (University of Stellenbosch)

Abstract: This presentation discusses a massively parallel framework for modular computations with polynomial data, leveraging the capabilities of the Singular/GPI-Space framework. Singular serves as the computational backbone, providing the algebraic engine, while GPI-Space manages the orchestration and communication of parallel tasks, relying on Petri nets as a mathematical modeling language. Convenient installation is realized through the Spack package manager. Our framework's strength lies in 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 will also discuss current work towards establishing a generic programming model that can further democratize access to massively parallel methods for commutative algebra and algebraic geometry, both for mathematians as well as applied scientists.

Massively parallel methods in birational geometry

Benjamin Mirgain (Universität des Saarlandes, Fraunhofer ITWM)

Abstract: Computational birational geometry is one of the key playing fields in an algorithmic approach to algebraic geometry, since birational maps are the fundamental way to relate algebraic varieties (or schemes). An important application is an algorithmic approach to the Minimal Model Program (MMP), which aims to classify algebraic varieties with mild singularities by finding simple birational models of such varieties in their birational equivalence class. The aim of this talk is to develop massively parallel methods to solve problems in birational geometry and the MMP. Making use of a representation of algebraic varieties in terms of charts and employing modular methods for problems in characteristic zero, allows for a massively parallel computational approach for handling both the varieties and rational maps between them. For the MMP we aim to implement and utilize a proof of the finite generation of the cannonical ring as developed by Lazić.

Algorithms for Gromov-Witten invariants of elliptic curves

Ali Traore (RPTU)

Abstract: We present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from just-in-time compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the Gromov-Witten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasi-modularity and homogeneity.

Recent development and advanced features of NeatIBP

Zihao Wu (Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Zhejiang, China)

Abstract: NeatIBP is a software generating small-size integration-by-parts (IBP) relations for Feynman integral computation. It is parallelized and uses algorithms based on computational algebraic geometry. In this talk, we present some recent developements including several advanced features of NeatIBP. This includes simplification of the solution of syzygy equations, interfaces with IBP reduction software, the spanning cuts method, and corresponding parallelization strategies.

Massively Parallel Methods for Free Resolutions

Santosh Gnawali (RPTU)

Abstract: This talk introduces an approach aimed at applying massively parallel methods for computing syzygies and free resolutions, with a primary focus on Schreyer's Resolution. Our method capitalizes on the inherent parallelism of the algorithm, primarily utilizing Petri nets, within the GPI-Space framework, as our key language for parallelism. GPI-Space  is a task-based workflow management system that employs Petri nets as its coordination layer, while the computation is carried out by the  computer algebra system Singular We present in detail the way the algorithm is modelled through a Petri net, explaining the coordination of tasks and data structures within the parallel computing environment. Although our implementation is still work-in-progress, preliminary results show promising advancements in efficiency. By exploiting the potential synergy between the computer algebra system Singular and the GPI-Space, we anticipate significant reductions in computational time for computing free resolutions.

Santiago Laplagne (Instituto de Cálculo - FCEyN - UBA, Buenos Aires)

A Combinatorial Approach for Computing Integral Bases

Abstract: We present a new approach for computing an integral basis of an algebraic function field of one variable in characteristic zero. Our approach uses combinatorial optimization, and it can be a good strategy when the curve has many branches at the singularity. Moreover, parallelization techniques can be used for splitting the given polynomial into branches and computing the local integral basis for the branches efficiently. Joint work with J. Boehm, W. Decker, S. Laplagne, G. Pfister.

Marcel Wittmann (RPTU)

Multivariate Partial Fraction Decomposition for Feynman Integrals

Abstract: We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas' multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, We observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ~100. We observe that our algorithm also works well for settings without a UT basis.

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