Algebraic structures of polynomial data: applications to galois theory and quantum codes

Abstract

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over Q, combining explicit resolvent constructions with modern machine-learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants J0,...,J4 obtained via binary transvections. For each polynomial, carefully designed resolvents are factored to determine its Galois group among the seven transitive subgroups of S7. Using this dataset, we train neuro-symbolic classifiers that integrate invariant-theoretic features with supervised learning, achieving improved accuracy in detecting rare solvable groups compared to coefficient-based and purely numerical baselines. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into the distribution of Galois groups under height and discriminant constraints.Building on the same polynomial data, we construct families of superelliptic curves on weighted projective spaces and derive algebraic-geometry and quantum error-correcting codes from them. We use a weighted projective framework to control genus, singularities, and rational points, and we introduce graded evaluation codes adapted to weighted embeddings. Within the CSS construction, we obtain criteria for self-orthogonality and establish refined Singleton-type bounds for the resulting quantum weighted algebraic-geometry (QWAG) codes. Numerical experiments over finite fields show how Galois-theoretic labels, weighted models, and graded evaluation spaces interact to produce quantum codes with competitive parameters and support a systematic, database-driven search for good classical and quantum codes.