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.

翻译：多项式方程组解集通常包含病态的奇异点。奇点消解是几何学中的一个基本过程，它用光滑点替换奇异点，同时保持解集其余部分不变。奇点消解并非唯一确定：通常的描述方式涉及反复执行一种称为"爆破"的基本操作，其复杂度高度依赖于特定选择。该过程可转化为两人博弈的多种变体，即所谓的"广中平佑博弈"，其中首位玩家的获胜策略提供了奇点消解问题的解决方案。本文提出了一种基于强化学习代理的新方法，用于求解广中平佑博弈以寻找最优奇点消解方案。在特定领域内，训练后的模型在多项式加法总次数上优于最先进的选择启发式算法，这验证了机器学习的最新进展在提升符号计算算法性能方面的潜力。