# Learning selection strategies in Buchberger's algorithm

arXiv: https://arxiv.org/abs/2005.01917 · HTML: https://mlsanity.com/paper/2005.01917/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2020.

## Built on by (most cited first, 20 of 20)

- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Is deep learning a useful tool for the pure mathematician?](https://mlsanity.com/paper/2304.12602/) (2023)
- [Cluster Algebras: Network Science and Machine Learning](https://mlsanity.com/paper/2203.13847/) (2022)
- [Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/2304.12154/) (2023)
- [Learning to Compute Gr"obner Bases](https://mlsanity.com/paper/2311.12904/) (2023)
- [Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms](https://mlsanity.com/paper/2505.23696/) (2025)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [An ML approach to resolution of singularities](https://mlsanity.com/paper/2307.00252/) (2023)
- [From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook](https://mlsanity.com/paper/2202.06086/) (2022)
- [Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD](https://mlsanity.com/paper/2401.13343/) (2024)
- [Learning proofs for the classification of nilpotent semigroups](https://mlsanity.com/paper/2106.03015/) (2021)
- [Learning a performance metric of Buchberger's algorithm](https://mlsanity.com/paper/2106.03676/) (2021)
- [Predicting the cardinality and maximum degree of a reduced Gr"obner basis](https://mlsanity.com/paper/2302.05364/) (2023)
- [Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula](https://mlsanity.com/paper/2306.16951/) (2023)
- [Constrained Neural Networks for Interpretable Heuristic Creation to Optimise Computer Algebra Systems](https://mlsanity.com/paper/2404.17508/) (2024)
- [HATSolver: Learning Groebner Bases with Hierarchical Attention Transformers](https://mlsanity.com/paper/2512.14722/) (2025)
- [SC-Square: Future Progress with Machine Learning?](https://mlsanity.com/paper/2209.04361/) (2022)
- [Learning to be Simple](https://mlsanity.com/paper/2312.05299/) (2023)
- [Learning Fast Monomial Orders for Gr"obner Basis Computations](https://mlsanity.com/paper/2602.02972/) (2026)
- [k-server-bench: Automating Potential Discovery for the k-Server Conjecture](https://mlsanity.com/paper/2604.07240/) (2026)

## Abstract (arXiv)

Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.