Massively parallel computations in algebraic geometry - not a contradiction

TL;DR

This paper presents a framework combining Singular and GPI-Space for massively parallel computations in algebraic geometry, enabling efficient solutions to complex problems like smoothness testing, GIT-fans, and tropicalizations.

Contribution

It introduces a novel infrastructure for parallel algebraic geometry computations, integrating existing tools to handle diverse complex problems efficiently.

Findings

Successfully determined smoothness of algebraic varieties

Computed GIT-fans in geometric invariant theory

Determined tropicalizations using the framework

Abstract

The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In this note, we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra.