Learning to Compute Gröbner Bases

TL;DR

The experiments show that the 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.

Abstract

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 notorious doubly exponential time complexity in the number of variables in the worst case. This paper is the first to address the learning of Gröbner basis computation with Transformers. The training requires many pairs of a polynomial system and the associated Gröbner basis, raising two novel algebraic problems: random generation of Gröbner bases and transforming them into non-Gröbner ones, termed as backward Gröbner 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 Gröbner bases, and Gröbner computation is learnable in a particular class.

Figures (1)

• Figure 1: Visual analysis of embedding vectors of numbers given by the proposed embedding. Embedding $c\in {\mathbb{R}}$ to $f_{\mathrm{E}}(c) \in {\mathbb{R}}^{D}$ from $c_{\min}$ to $c_{\max}$ with $B$ bins to obtain $M \in {\mathbb{R}}^{B\times D}$, the fix figures show from the left, (i) the Euclidean distance matrix of $M$, (ii) its slice at $0$, (iii) the norm of embedding vectors, (iv) the dot product $\tilde{M}\tilde{M}^{\top}$ with $\tilde{M}$ of the row-normalized $M$, (v) $f_{\mathrm{E}}(0)^{\top}\tilde{M}$ and (vi) $f_{\mathrm{E}}(c_0)^{\top}M$. (a) Trained on ${\mathbb{R}}[x_1, x_2]$; $(c_{\min}, c_{\max}) = (-100, 100)$. (b) Trained on ${\mathbb{F}}_{31}[x_1, x_2]$; $(c_{\min}, c_{\max}) = (0, 31)$. The embedding layer $f_{\mathrm{E}}$ has one/two hidden layers (top/bottom rows). As can be seen, the relationship between embedding vectors in terms of distance and dot product is aligned well in the infinite field and not in the finite field.

Theorems & Definitions (30)

• Definition 3.1: Leading term
• Definition 3.2: Gröbner basis
• Definition 4.3: 0-dimensional ideal
• Definition 4.4: Shape position
• Theorem 4.6
• Proposition 4.6
• Theorem 4.7
• Definition A.1: Ring, Field (atiyah1994introduction, Chap. 1 §1)
• Definition A.2: Polynomial Ring (atiyah1994introduction, Chap. 1 §1)
• Definition A.3: Quotient Ring (atiyah1994introduction, Chap. 1 §1)