Learning to Compute Gr\"obner Bases

Abstract

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 notorious doubly exponential time complexity in the number of variables in the worst case. This paper is the first to address the learning of Gr\"obner basis computation with Transformers. The training requires many pairs of a polynomial system and the associated Gr\"obner basis, raising two novel algebraic problems: random generation of Gr\"obner bases and transforming them into non-Gr\"obner ones, termed as backward Gr\"obner problem. We resolve these problems with 0-dimensional radical ideals, the ideals appearing in various applications. Further, we propose a hybrid input embedding to handle coefficient tokens with continuity bias and avoid the growth of the vocabulary set. The experiments show that our dataset generation method is a few orders of magnitude faster than a naive approach, overcoming a crucial challenge in learning to compute Gr\"obner bases, and Gr\"obner computation is learnable in a particular class.