Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notoriously expensive computational cost -- doubly exponential time complexity in the number of variables in the worst case. In this paper, we achieve for the first time Gr\"obner basis computation through the training of a transformer. The training requires many pairs of a polynomial system and the associated Gr\"obner basis, thus motivating us to address two novel algebraic problems: random generation of Gr\"obner bases and the transformation of them into non-Gr\"obner polynomial systems, termed as \textit{backward Gr\"obner problem}. We resolve these problems with zero-dimensional radical ideals, the ideals appearing in various applications. The experiments show that in the five-variate case, the proposed dataset generation method is five orders of magnitude faster than a naive approach, overcoming a crucial challenge in learning to compute Gr\"obner bases.

通过训练转化器实现了首次Gröbner基的计算，该训练需要多个多项式系统和相关Gröbner基的配对，从而引发了两个新的代数问题：Gröbner基的随机生成和将其转化为非Gröbner多项式系统，称为“反向Gröbner问题”。我们通过零维根理想解决了这些问题，并且实验证明，在五元情况下，所提出的数据集生成方法比朴素方法快了五个数量级，克服了在学习计算Gröbner基中的重要挑战。

学习计算格罗布纳基