Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials

2025-02-28

AI Summary

This work addresses the accuracy and interpretability bottlenecks in classifying Galois groups of irreducible sextic polynomials. Method: We propose a neuro-symbolic model that integrates neural networks for learning coefficient-based features and performing Galois group classification, with symbolic computation to generate invariants, verify group structures, and analyze distributional regularities under algebraic number-theoretic constraints. Contribution/Results: We present the first systematic empirical characterization of Galois group probability distributions across 53,972 irreducible sextic polynomials. We discover that the cyclic group (C_6) is fully covered by only seven invariant equivalence classes—providing the first empirical foundation for conjectures on Galois group probabilities and solvability by radicals. Furthermore, the observed compression规律 of equivalence classes substantially enhances both classification accuracy and interpretability, outperforming purely numerical approaches.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Classify Galois groups of polynomials using neurosymbolic methods.
• Analyze distributional trends in sextic polynomials with height constraints.
• Explore solvability by radicals and probabilistic conjectures in algebra.

Innovation

• Neuro-symbolic approach for Galois group classification
• Neural networks combined with symbolic reasoning
• Analysis of sextic polynomials with height constraints