Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Elira Shaska, Tony Shaska

TL;DR

This work tackles the problem of classifying Galois groups $G = \mathrm{Gal}_{\mathbb{Q}}(f)$ of polynomials by uniting neural pattern recognition with algebraic reasoning in a neuro-symbolic framework. Focusing on sextics with height $\leq 6$, the authors build a dataset of $53{,}972$ irreducible polynomials and integrate invariants, signatures, discriminants, and resolvents into symbolic layers that guide learning. The resulting $\mathrm{GaloisNetwork}$ achieves improved accuracy and interpretability over purely numerical methods, and uncovers distributional patterns, such as $20$ polynomials with $\mathrm{Gal}(f) \cong C_6$ distributed across seven $\mathrm{GL}_2(\mathbb{Q})$-equivalence classes, offering empirical insight into probabilistic Galois theory under height constraints. The work lays a foundation for AI-assisted computational algebra, with open data and pathways to higher-degree classification and solvability-by-radicals.

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.

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

TL;DR

This work tackles the problem of classifying Galois groups of polynomials by uniting neural pattern recognition with algebraic reasoning in a neuro-symbolic framework. Focusing on sextics with height , the authors build a dataset of irreducible polynomials and integrate invariants, signatures, discriminants, and resolvents into symbolic layers that guide learning. The resulting achieves improved accuracy and interpretability over purely numerical methods, and uncovers distributional patterns, such as polynomials with distributed across seven -equivalence classes, offering empirical insight into probabilistic Galois theory under height constraints. The work lays a foundation for AI-assisted computational algebra, with open data and pathways to higher-degree classification and solvability-by-radicals.

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 , we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group 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.

Paper Structure

This paper contains 14 sections, 4 theorems, 6 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Galois groups of polynomials
  3. Reduction modulo a prime and signature of groups
  4. Polynomials with non-real roots
  5. A word on equivalence classes of polynomials
  6. Resolvents
  7. Probabilistic Galois Theory
  8. Irreducible sextics: a case study
  9. Counting polynomials
  10. Data quality versus data quantity
  11. Neurosymbolic Networks for Galois Group Determination
  12. Symbolic Layers: Encoding Mathematical Knowledge
  13. The Galois Network: A Neurosymbolic Architecture
  14. Conclusions

Key Result

Theorem 2.1

(Dedekind) Let $f(x) \in {\mathbb Z}[x]$ be a monic polynomial such that $\deg f = n$, $\mathop{\mathrm{Gal }}\nolimits_{\mathbb Q} (f) = G$, and $p$ a prime such that $p \nmid {\Delta}_f$. If $f_p:=f(x) \mod p \, \,$ factors in ${\mathbb F}_p [x]$ as a product of irreducible factors of degree $n_1,

Figures (1)

  • Figure 1: Lattice of transitive subgroups of $S_6$ with blue nodes representing subgroups of $A_6$.

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2: 2004-1
  • Theorem 2.3
  • Lemma 1