A Neurosymbolic Framework for Geometric Reduction of Binary Forms
This work addresses the reduction of binary forms in symbolic computation, offering incremental improvements and a hybrid AI approach for researchers in algebra and geometry.
The paper tackled the problem of finding equivalent binary forms with minimal coefficients by comparing Julia and hyperbolic reduction, showing that hyperbolic reduction generally outperforms, especially for sextics and decimics, though neither guarantees minimality. It proposed a shift and scaling to better approximate minimal forms and introduced a machine learning framework to optimize transformations for minimizing heights.
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.