Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials
This work addresses challenges in algebraic computation for researchers in computational algebra, offering incremental advances by applying a neuro-symbolic method to a specific domain.
This paper tackled the problem of classifying Galois groups of polynomials by integrating neural networks with symbolic reasoning, achieving improved accuracy and interpretability over numerical methods on a dataset of 53,972 irreducible sextic polynomials, uncovering novel distributional trends such as 20 polynomials with Galois group C6 spanning seven equivalence classes.
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.