Embedded desingularization for arithmetic surfaces -- toward a parallel implementation

TL;DR

This paper introduces a parallel algorithm for embedded desingularization of arithmetic surfaces, emphasizing implementability and efficiency in mixed characteristic cases, utilizing Petri nets for parallel execution.

Contribution

It presents a novel variant of desingularization algorithm that refines complexity measures and adapts to parallel computation environments.

Findings

Algorithm successfully handles mixed characteristic cases.

Parallel implementation improves efficiency and scalability.

Refinement of complexity measure enhances desingularization process.

Abstract

We present an algorithmic embedded desingularization of arithmetic surfaces bearing in mind implementability. Our algorithm is based on work by Cossart-Jannsen-Saito, though our variant uses a refinement of the order instead of the Hilbert-Samuel function as a measure for the complexity of the singularity. We particularly focus on aspects arising when working in mixed characteristics. Furthermore, we exploit the algorithm's natural parallel structure rephrasing it in terms of Petri nets for use in the parallelization environment GPI-Space with {\sc Singular} as computational back-end.