An ML approach to resolution of singularities

TL;DR

This paper introduces a machine learning approach using reinforcement learning agents to improve the process of resolving singularities in polynomial systems, outperforming traditional heuristics in some cases.

Contribution

It presents a novel ML-based method for the Hironaka game, demonstrating potential to enhance symbolic computation algorithms.

Findings

Reinforcement learning agents can effectively solve the Hironaka game.

The ML approach outperforms existing heuristics in polynomial addition efficiency.

Proof-of-concept shows ML can improve symbolic computation processes.

Abstract

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.