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：多项式方程组的解集通常包含不良的奇异点。奇点解消是几何中的一个基本过程，我们在此过程中用光滑点替换奇异点，同时保持解集的其余部分不变。解消不具有唯一性：其通常的描述方式涉及重复执行一种称为"blow-up"的基本操作，而解消的复杂性高度依赖于某些选择。该过程可以转化为不同版本的二人博弈，即所谓的"Hironaka博弈"，其中第一玩家的必胜策略为奇点解消问题提供了解决方案。本文提出了一种利用强化学习智能体寻找最优奇点解消的Hironaka博弈新方法。在特定领域内，训练好的模型在执行的多项式加法总数上超越了最先进的选择启发式方法，这证明了机器学习的最新进展有潜力提升符号计算算法的性能。