用神经符号网络分类多项式伽罗瓦群，提升求解效率。
Galois groups of polynomials and neurosymbolic networks
- 设计神经符号网络识别多项式的伽罗瓦群
- 发现非对称/交错群的多项式分布规律
- 相比传统神经网络更高效，适合代数研究者
本文提出一种新方法，通过机器学习理解伽罗瓦理论这一代数学核心领域。通过分析多项式方程，旨在简化根式可解性的判定过程，并拓展其在伽罗瓦理论中的应用。具体地，我们设计了一种神经符号网络用于伽罗瓦群分类，结果显示其效率优于传统神经网络。同时，我们发现了非对称群与交错群同构的伽罗瓦群所对应的多项式具有特殊分布特征。该工作展示了数据科学在代数领域的潜力，同时也指出了方法面临的挑战。
原文摘要 · Abstract (English)
This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline the process of determining solvability by radicals and explore broader applications within Galois theory. This summary encapsulates the background, methodology, potential applications, and challenges of using data science in Galois theory. More specifically, we design a neurosymbolic network to classify Galois groups and show how this is more efficient than usual neural networks. We discover some very interesting distribution of polynomials for groups not isomorphic to the symmetric groups and alternating groups.
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