构建百万级多项式数据库，用机器学习识别七次方程的伽罗瓦群。
From Polynomials to Databases: Arithmetic Structures in Galois Theory
- 结合显式预解式与神经符号分类器，分析七次多项式的伽罗瓦群。
- 在超过一百万条数据上验证，对稀有可解群的识别准确率显著提升。
- 适合代数计算、数论研究者，尤其关注伽罗瓦理论与自动化推理者。
我们构建了一个计算框架，用于分类定义在有理数域上的不可约七次多项式的伽罗瓦群，结合显式预解式方法与机器学习技术。创建了一个包含超过一百万条归一化射影七次多项式的数据库，每条均标注由二元扭转导出的代数不变量 $J_0, \dots, J_4$。对每个多项式，通过计算预解式分解来确定其伽罗瓦群，从 $S_7$ 的七个传递子群中识别（由 Foulkes 确定）。利用该数据集，训练了一种融合不变量理论特征与监督学习的神经符号分类器，在检测稀有可解群方面优于基于系数的模型。所得数据库为构造性伽罗瓦理论提供了可复现资源，并支持在高度约束下对群分布的实证研究。该方法可扩展至更高次数情形，展示了混合符号-数值技术在计算代数中的潜力。
原文摘要 · Abstract (English)
We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ identified by Foulkes. Using this dataset, we train a neurosymbolic classifier that integrates invariant-theoretic features with supervised learning, yielding improved accuracy in detecting rare solvable groups compared to coefficient-based models. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into group distribution under height constraints. The methodology extends to higher-degree cases and illustrates the utility of hybrid symbolic-numeric techniques in computational algebra.
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