Neuro-Symbolic Learning for Irreducible Sextics: Unveiling Probabilistic Trends in Polynomials

Elira Shaska, Tanush Shaska

I-AIMS paper sh-86 · 2025

This paper presents a neuro-symbolic 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—leveraging invariants like root distributions, signatures, and resolvents—we develop a model that outperforms purely numerical methods in accuracy and interpretability. Focusing on sextic polynomials with height ≤ 6, we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group C₆ 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.

MSC classes: Primary 12F10; Secondary 68T07

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