An ML approach to resolution of singularities
This work addresses a fundamental challenge in symbolic computation for mathematicians and computer scientists, offering an incremental improvement through machine learning integration.
The paper tackles the problem of resolving singularities in polynomial systems by introducing a reinforcement learning approach to the Hironaka game, which outperforms state-of-the-art heuristics in reducing the number of polynomial additions in certain domains.
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.