Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms

Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high computational costs, which have motivated recent Deep Learning approaches to improve efficiency, albeit at the expense of output correctness. In this work, we introduce the Oracle Border Basis Algorithm, the first Deep Learning approach that accelerates Border basis computation while maintaining output guarantees. To this end, we design and train a Transformer-based oracle that identifies and eliminates computationally expensive reduction steps, which we find to dominate the algorithm's runtime. By selectively invoking this oracle during critical phases of computation, we achieve substantial speedup factors of up to 3.5x compared to the base algorithm, without compromising the correctness of results. To generate the training data, we develop a sampling method and provide the first sampling theorem for border bases. We construct a tokenization and embedding scheme tailored to monomial-centered algebraic computations, resulting in a compact and expressive input representation, which reduces the number of tokens to encode an $n$-variate polynomial by a factor of $O(n)$. Our learning approach is data efficient, stable, and a practical enhancement to traditional computer algebra algorithms and symbolic computation.

Learning to Compute Gröbner Bases

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.