# An ML approach to resolution of singularities

arXiv: https://arxiv.org/abs/2307.00252 · HTML: https://mlsanity.com/paper/2307.00252/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 29 references that are in the map.
- First posted on arXiv 1 Jul 2023.

## Builds on (most cited first, 2 of 2)

- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [Learning selection strategies in Buchberger's algorithm](https://mlsanity.com/paper/2005.01917/) (2020)

## Built on by (most cited first, 3 of 3)

- [Flow-based Extremal Mathematical Structure Discovery](https://mlsanity.com/paper/2601.18005/) (2026)
- [Algebraic Representations for Faster Predictions in Convolutional Neural Networks](https://mlsanity.com/paper/2408.07815/) (2024)
- [Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic](https://mlsanity.com/paper/2602.06553/) (2026)

## Abstract (arXiv)

The solution set of a system of polynomial equations typically contains ill-behaved, singular points. Resolution is a fundamental process in geometry in which we replace singular points with smooth points, while keeping the rest of the solution set unchanged. Resolutions are not unique: the usual way to describe them involves repeatedly performing a fundamental operation known as "blowing-up", and the complexity of the resolution highly depends on certain choices. The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. 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.