超越F5和GVW：用于快速理想基计算的恰当覆盖算法
Beyond F5 and GVW: The Proper-Cover Algorithm for Fast Ideal Basis Computation
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中文总结 AI 辅助
研究零维多项式理想的快速计算方法，结合GVW覆盖优化与恰当基理论提出恰当覆盖算法，推广相关概念设计两阶段算法，证明其性质，该算法在多种单项式排序下性能优于F5和GVW。
中文摘要 AI 辅助
格罗贝格基计算会产生大量计算开销，尤其是在字典序下。F5及其GVW变体主导了基于域的高效格罗贝格基求解。恰当基算法提供了一个参数化理想计算框架，且未利用现代基于签名的优化。本文通过将GVW对签名的覆盖优化与恰当基理论相结合，提出了用于零维多项式理想的恰当覆盖算法。我们将签名、覆盖、POT排序、约化和S对概念推广到参数化系数，设计了具有兼容因子构造和饥饿细化的两阶段算法，并严格证明了终止性和输出正确性。基准测试结果表明，恰当覆盖在所有单项式排序下都优于F5，在字典序下比GVW有明显加速。
英文摘要
Gröbner basis computation incurs heavy computational overhead, especially under lexicographic order. F5 and its GVW variant dominate efficient field-based Gröbner basis solving. The proper basis algorithm offers a parameterized ideal computation framework without leveraging modern signature-based optimizations. This work presents the Proper-Cover algorithm for zero-dimensional polynomial ideals by combining GVW's cover optimization over signature with the proper basis theory. We generalize signature, cover, POT ordering, reduction and S-pair concepts to parameterized coefficients, design a two-phase algorithm with compatible factor construction and hungry refinement, and rigorously prove termination and output correctness. Accordingly, we propose a new framework for the efficient computation of polynomial ideal bases. Benchmark results show that Proper-Cover surpasses F5 under all monomial orderings and delivers clear speedups over GVW for lexicographic (plex) order.
