Primary decomposition of modules: a computational differential approach

TL;DR

This paper develops a differential and computational framework for primary decomposition of modules over polynomial rings, introducing a structure theory, characterization via differential operators, and an algorithm for minimal differential primary decomposition.

Contribution

It provides a new structure theory and a computational algorithm for primary decomposition of modules using differential operators.

Findings

Established a general structure theory for primary submodules.

Characterized primary submodules in terms of differential operators.

Implemented an algorithm for minimal differential primary decomposition.

Abstract

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.