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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Published in:	Foundations of computational mathematics Vol. 21; no. 3; pp. 767 - 806
Main Authors:	Böhm, Janko, Decker, Wolfram, Frühbis-Krüger, Anne, Pfreundt, Franz-Josef, Rahn, Mirko, Ristau, Lukas
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.06.2021 Springer Nature B.V
Subjects:	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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Abstract	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.
AbstractList	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. 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.
Author	Decker, Wolfram Böhm, Janko Rahn, Mirko Frühbis-Krüger, Anne Pfreundt, Franz-Josef Ristau, Lukas
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Cites_doi	10.1007/978-3-662-29244-0 10.1016/S0022-0000(69)80011-5 10.1093/comjnl/32.2.98 10.1007/978-3-319-70566-8_3 10.1145/129630.129635 10.1016/j.jsc.2017.05.001 10.1017/CBO9780511608681 10.1007/978-3-642-57739-0 10.1007/s00287-006-0107-7 10.24033/asens.1286 10.1007/3-540-11607-9_18 10.1007/s002220050141 10.1145/359576.359579 10.1007/978-1-4615-5499-8_10 10.4171/RMI/425 10.2140/jsag.2012.4.6 10.1007/978-3-662-10427-9_6 10.1177/1094342010391989 10.1090/S0002-9947-09-04716-3 10.1007/978-3-642-20300-8_1 10.1016/0001-8708(82)90048-2 10.2307/1970547 10.1016/0166-218X(91)90114-C 10.1142/9789812706812_0008 10.24033/asens.1573 10.1145/800070.802201 10.1007/PL00012443 10.1016/0304-3975(92)90173-D 10.1145/800076.802477 10.1016/j.jsc.2012.07.002 10.1007/978-1-4612-5350-1 10.1007/978-3-662-06289-0_1 10.1090/mcom/2951 10.1142/S0218196715500332 10.1007/BF01390081 10.1016/0021-8693(83)90096-0
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References_xml	– reference: Hauser, H., Why the characteristic zero proof of resolution of singularities fails in positive characteristic, Manuscript (2003), https://homepage.univie.ac.at/herwig.hauser/ – reference: Böhm J.; Papadakis S., KustinMiller – The Kustin-Miller complex construction and resolutions of Gorenstein rings, Macaulay2 package (2012). – reference: Jensen, K., Coloured Petri Nets. Volume 1. Springer (1992). – reference: BöhmJFrühbis-KrügerAA smoothness test for higher codimensionsJ. Symbolic Comput.201886153165372521810.1016/j.jsc.2017.05.001 – reference: EncinasSHauserHStrong resolution of singularities in characteristic zeroComment. Math. Helv.200277821845194911510.1007/PL00012443 – reference: BöhmJDeckerWFiekerCPfisterGThe use of bad primes in rational reconstructionMath. 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Snippet	Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on...
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SubjectTerms	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
Title	Towards Massively Parallel Computations in Algebraic Geometry
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