Solving a polynomial system, or computing an associated Gröbner 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 a recent article soon to appear, we achieve for the first time Gröbner basis computation through the training of a transformer. The training requires many pairs of a polynomial system and the associated Gröbner basis, thus motivating us to address two novel algebraic problems: random generation of Gröbner bases and the transformation of them into non-Gröbner polynomial systems, termed as backward Gröbner 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 Gröbner bases.
