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

Abstract

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.