Towards Massively Parallel Computations in Algebraic Geometry

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 hav...

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Veröffentlicht in:Foundations of computational mathematics Jg. 21; H. 3; S. 767 - 806
Hauptverfasser: Böhm, Janko, Decker, Wolfram, Frühbis-Krüger, Anne, Pfreundt, Franz-Josef, Rahn, Mirko, Ristau, Lukas
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.06.2021
Springer Nature B.V
Schlagworte:
Algebra
Applications of Mathematics
Computer algebra
Computer models
Computer networks
Computer Science
Coordination
Data processing
Distributed processing
Economics
Geometry
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Petri nets
Polynomials
Programming languages
Smoothness
Workflow management systems
Hironaka desingularization
68W30
Surfaces of general type
Petri nets
14Q99
68W10
Computational algebraic geometry
Smoothness test
Distributed computing
14B05
Singular
GPI-Space
Computer algebra
ISSN:1615-3375, 1615-3383
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Zusammenfassung: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-020-09464-x