A Neurosymbolic Framework for Geometric Reduction of Binary Forms

TL;DR

This paper compares reduction methods for binary forms, introduces a machine learning approach to optimize transformations, and demonstrates potential improvements in minimizing form coefficients through hybrid symbolic and data-driven techniques.

Contribution

It presents a comparative analysis of Julia and hyperbolic reduction, proposes an enhanced reduction process, and introduces a machine learning framework for optimal binary form transformations.

Findings

Hyperbolic reduction outperforms Julia reduction for sextics and decimics.

Additional shift and scaling improve approximation to minimal forms.

Machine learning effectively identifies transformations that minimize form heights.

Abstract

This paper compares Julia reduction and hyperbolic reduction with the aim of finding equivalent binary forms with minimal coefficients. We demonstrate that hyperbolic reduction generally outperforms Julia reduction, particularly in the cases of sextics and decimics, though neither method guarantees achieving the minimal form. We further propose an additional shift and scaling to approximate the minimal form more closely. Finally, we introduce a machine learning framework to identify optimal transformations that minimize the heights of binary forms. This study provides new insights into the geometry and algebra of binary forms and highlights the potential of AI in advancing symbolic computation and reduction techniques. The findings, supported by extensive computational experiments, lay the groundwork for hybrid approaches that integrate traditional reduction methods with data-driven techniques.