Massively Parallel Modular Methods in Commutative Algebra and Algebraic Geometry

TL;DR

This work presents a novel, massively parallel framework for modular computations with polynomial data, able to cover a broad spectrum of applications in commutative algebra and algebraic geometry, and illustrates that the approach scales very well with the number of processor cores used for the computation.

Abstract

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. 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 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.

Figures (10)

• Figure 1: Task and data parallelism in a Petri net.
• Figure 2: Modular algorithm Petri net
• Figure 3: Subnet Lift
• Figure 4: Subnet Lift/Add
• Figure 5: Subnet ReconsTest

Theorems & Definitions (95)

• Lemma 2.1.1
• Remark 2.1.2
• Theorem 2.1.3
• Remark 2.1.4
• Example 2.1.5
• Definition 2.2.1
• Definition 2.2.2
• Definition 2.2.3
• Definition 2.2.4
• Definition 2.2.5