LipschitzSaturation: A Macaulay2 Package for Computing Lipschitz Saturations of Modules and Toric Varieties

We introduce \verb|LipschitzSaturation|, a package for the computer algebra system $\mathit{\text{Macaulay2}}$ 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 $\mathit{\text{Macaulay2}}$ version 1.22 or later.