A data structure for monomial ideals with applications to signature Gröbner bases

2026-01-08

arXiv.org

AI Summary

This work proposes the Monomial Divisibility Diagram (MDD), a directed acyclic graph structure based on shared subtrees, to accelerate zero-reduction detection in symbolic Gröbner basis computations. The MDD efficiently represents monomial ideals and supports fast generator insertion and membership testing. By transforming the canonical tree representation with maximally shared subtrees into a practical data structure, the MDD integrates tree normalization, hash-based node sharing, and algebraic algorithmic components to significantly enhance operational efficiency. Integration of the MDD into AlgebraicSolving.jl yields substantial speedups in zero-reduction detection, demonstrating its effectiveness in improving the performance of symbolic computation workflows.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We introduce monomial divisibility diagrams (MDDs), a data structure for monomial ideals that supports insertion of new generators and fast membership tests. MDDs stem from a canonical tree representation by maximally sharing equal subtrees, yielding a directed acyclic graph. We establish basic complexity bounds for membership and insertion, and study empirically the size of MDDs. As an application, we integrate MDDs into the signature Gr\"obner basis implementation of the Julia package AlgebraicSolving.jl. Membership tests in monomial ideals are used to detect some reductions to zero, and the use of MDDs leads to substantial speed-ups.

Problem

• monomial ideals
• data structure
• membership test
• signature Gröbner bases
• zero reduction

Innovation

• monomial ideals
• directed acyclic graph
• Gröbner bases
• data structure
• membership test