TY - JOUR

T1 - Towards Massively Parallel Computations in Algebraic Geometry

AU - Böhm, Janko

AU - Decker, Wolfram

AU - Frühbis-Krüger, Anne

AU - Pfreundt, Franz-Josef

AU - Rahn, Mirko

AU - Ristau, Lukas

N1 - Publisher Copyright: © 2020, SFoCM.

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

KW - Computational algebraic geometry

KW - Computer algebra

KW - Distributed computing

KW - GPI-Space

KW - Hironaka desingularization

KW - Petri nets

KW - Singular

KW - Smoothness test

KW - Surfaces of general type

UR - http://www.scopus.com/inward/record.url?scp=85087557425&partnerID=8YFLogxK

U2 - 10.1007/s10208-020-09464-x

DO - 10.1007/s10208-020-09464-x

M3 - Article

VL - 21

SP - 767

EP - 806

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 3

ER -