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

Learning to Compute Gröbner Bases

We describe a parametrized space for simple meta-reinforcement-learning
(meta-RL) tasks with arbitrary stimuli. The parametrization allows us to
randomly generate an arbitrary number of novel simple meta-learning tasks. The
space of meta-RL tasks covered by this parametrization includes many well-known
meta-RL tasks, such as bandit tasks, the Harlow task, T-mazes, the Daw two-step
task and others. Simple extensions allow it to capture tasks based on
two-dimensional topological spaces, such as find-the-spot or key-door tasks. We
describe a number of randomly generated meta-RL tasks and discuss potential
issues arising from random generation.

利用参数化空间随机生成简单元元强化学习任务，覆盖许多已知的任务，并能对二维拓扑任务进行扩展。作者描述了几个随机生成的元元强化学习任务，并讨论相关问题。