Galois groups of polynomials and neurosymbolic networks

2025-01-22

AI Summary

This work addresses the longstanding challenges of automatic Galois group classification and solvability-by-radicals determination. We propose the first neural-symbolic network architecture specifically designed for Galois theory. Methodologically, it integrates polynomial algebraic feature engineering, group-theoretic constraint embedding, symbolic reasoning guidance, and end-to-end deep learning training—thereby unifying structural prior knowledge with data-driven learning. Compared to purely data-driven models, our approach achieves a +12.7% accuracy gain on Galois group classification while significantly enhancing model interpretability. Furthermore, we uncover, for the first time, statistically skewed distribution patterns of polynomials whose Galois groups have order ≤ 12; we also identify nontrivial clustering of nonsolvable polynomials associated with non-symmetric and non-alternating groups. These findings establish a novel paradigm for machine learning–driven modeling of algebraic structures.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Machine Learning
• Galois Theory
• Neural-Symbolic Networks

Innovation

• Neuro-Symbolic Networks
• Galois Theory
• Mathematical Pattern Discovery