Lipschitz saturation of toric singularities in any dimension

TL;DR

This paper characterizes the Lipschitz saturation of complex toric singularities across all dimensions, providing criteria, algorithms, and comparisons with existing notions.

Contribution

It introduces a necessary and sufficient condition for monomials to belong to the Lipschitz saturation, along with a finite computation algorithm.

Findings

Provides a criterion based on Newton polyhedra and lattice conditions.

Establishes a finite algorithm for computing Lipschitz saturation.

Shows that presaturation differs from Lipschitz saturation in higher dimensions.

Abstract

We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.