用神经符号方法分析多项式伽罗瓦群，发现新概率规律。
Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials
- 结合神经网络与符号推理，提升分类准确率与可解释性。
- 在高度≤6的53972个六次不可约多项式中发现分布趋势。
- 适合对计算代数、机器学习交叉研究感兴趣的人阅读。
本文提出一种神经符号方法，用于分类多项式的伽罗瓦群，融合经典伽罗瓦理论与机器学习以应对代数计算挑战。通过将神经网络与符号推理结合，所提模型在准确率和可解释性上优于纯数值方法。聚焦于高度≤6的六次多项式，分析了包含53,972个不可约例子的数据库，揭示了新的分布规律，例如仅有20个六次多项式的伽罗瓦群为$C_6$，却仅分布在七个由不变量定义的等价类中。这些发现首次提供了在高度约束下伽罗瓦群概率的实证洞察，并为可根式求解性研究奠定基础。该工作展示了人工智能揭示传统符号技术难以发现模式的潜力，为计算代数的未来研究铺平道路，具有探索概率猜想与高次分类的深远意义。
原文摘要 · Abstract (English)
This paper presents a neurosymbolic approach to classifying Galois groups of polynomials, integrating classical Galois theory with machine learning to address challenges in algebraic computation. By combining neural networks with symbolic reasoning we develop a model that outperforms purely numerical methods in accuracy and interpretability. Focusing on sextic polynomials with height $\leq 6$, we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group $C_6$ spanning just seven invariant-defined equivalence classes. These findings offer the first empirical insights into Galois group probabilities under height constraints and lay the groundwork for exploring solvability by radicals. Demonstrating AI's potential to reveal patterns beyond traditional symbolic techniques, this work paves the way for future research in computational algebra, with implications for probabilistic conjectures and higher degree classifications.
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