Lipschitz饱和：一个用于计算模和环面簇的Lipschitz饱和的Macaulay2软件包
LipschitzSaturation: A Macaulay2 Package for Computing Lipschitz Saturations of Modules and Toric Varieties
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中文总结 AI 辅助
介绍用于Macaulay2的LipschitzSaturation软件包，它能计算模和环面簇的Lipschitz饱和。在模设置中处理多种饱和概念，通过基于曲线的成员测试解决计算难题，还实现了环面奇点构造算法，该软件包免费且需特定版本的Macaulay2。
中文摘要 AI 辅助
我们介绍了LipschitzSaturation，这是一个用于计算机代数系统Macaulay2的软件包，它实现了计算模和环面簇的Lipschitz饱和的算法。在模设置中，该软件包处理$\mathcal{O}_{X}$-子模$\mathcal{M}\subseteq\mathcal{O}_{X}^{p}$的三种不同饱和概念：1-、2-和3-Lipschitz饱和$\mathcal{M}_{S_{1}}$、$\mathcal{M}_{S_{2}}$和$\mathcal{M}_{S_{3}}$，以及双模$\mathcal{M}_{D}$的辅助构造。为绕过$\mathcal{M}_{S_{1}}$计算上棘手的多变量计算，实现了基于曲线的成员测试，在导致纯代数方法超时或耗尽内存的参数族上实现了近恒定运行时间。对于环面奇点，实现了一种构造算法。该软件包免费可用，需要Macaulay2版本1.22或更高版本。
英文摘要
We introduce \verb|LipschitzSaturation|, a package for the computer algebra system \textit{Macaulay2} that implements algorithms for computing Lipschitz saturations of modules and toric varieties. In the module setting, the package handles three distinct saturation notions for an $\mathcal{O}_{X}$-submodule $\mathcal{M}\subseteq\mathcal{O}_{X}^{p}$: the 1-, 2-, and 3-Lipschitz saturations $\mathcal{M}_{S_{1}}$, $\mathcal{M}_{S_{2}}$ and $\mathcal{M}_{S_{3}}$, together with the auxiliary construction of the double module $\mathcal{M}_{D}$. To bypass the computationally intractable multivariate calculations for $\mathcal{M}_{S_{1}}$, we implemented a curve-based membership test, achieving near-constant runtime on parametric families that cause the purely algebraic method to time out or exhaust memory. For Toric Singularities, we implemented a construction algorithm. The package is freely available and requires \textit{Macaulay2} version 1.22 or later.
