用AI优化二元型化简，提升计算效率与精度。
A Neurosymbolic Framework for Geometric Reduction of Binary Forms
- 结合几何与符号方法，改进传统化简算法。
- 在六次和十次型中，双曲化简优于朱利亚化简。
- 引入机器学习识别最优变换，适合符号计算研究者。
本文比较了朱利亚化简与双曲化简方法，旨在寻找系数最小的等价二元型。实验表明，双曲化简在六次型和十次型中表现更优，但两者均无法保证达到最小形式。为此，我们提出额外的平移与缩放操作以更接近最小形式。最后，引入机器学习框架，用于识别使二元型高度最小化的最优变换。研究通过大量计算实验验证，揭示了二元型的代数与几何特性，并展示了人工智能在推动符号计算与化简技术中的潜力。成果为传统化简方法与数据驱动技术融合的混合方法奠定了基础。
原文摘要 · Abstract (English)
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.
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